The Concepts of the Calculus: A Critical and Historical Discussion of the Derivative and the Integral 9780231892728

Table of contents :
Preface
Contents
I. Introduction
II. Conceptions in Antiquity
III. Medieval Contributions
IV. A Century of Anticipation
V. Newton and Leibniz
VI. The Period of Indecision
VII. The Rigorous Formulation
VIII. Conclusion
Bibliography
Index

Citation preview

The Concepts of the Calculus

The Concepts of the Calculus A CRITICAL AND HISTORICAL DISCUSSION OF THE DERIVATIVE AND THE INTEGRAL

By CARL B. BOYER

New York: Morningside Heights COLUMBIA UNIVERSITY PRESS 1939

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Preface

S

OME ten years ago Professor Frederick Barry, of Columbia University, pointed out to me that the history of the calculus had not been satisfactorily written. Other duties and inadequate preparation at the time made it impossible to act upon this suggestion, but my studies of the past several years have confirmed this view. There is indeed no lack of material on the origin and subject matter of the calculus, as the titles in the bibliography appended to this work will attest. What is wanting is a satisfactory critical account of the filiation of the fundamental ideas of the subject from their incipiency in antiquity to the final formulation of these in the precise concepts familiar to every student of the elements of modern mathematical analysis. The present work is an attempt to supply, in some measure, this deficiency. An authoritative and comprehensive treatment of the whole history of the elementary calculus is greatly to be desired; but any such ambitious project is far beyond the scope and intention of the dissertation here presented. This is not a history of the calculus in all its aspects, but a suggestive outline of the development of the basic concepts, and as such should be of service both to students of mathematics and to scholars in the field of the history of thought. The aim throughout has therefore been to secure clarity of exposition, rather than to present a confusingly elaborate all-inclusiveness of detail or to display a meticulously precise erudition. This has necessitated a judicious selection and presentation of such material as would preserve the continuity of thought, but it is to be hoped that historical accuracy and perspective have not thereby been sacrificed.

The inclusion at the end of this volume of an extensive bibliography of works to which reference has been made has caused it to appear unnecessary to give full citations in the footnotes. In these notes author and title—in some cases abbreviated—alone have been given; titles of books have been italicized, those of articles in the periodical literature appear in Roman type enclosed within quotation marks. It is felt that the inclusion of such a list of sources on the subject may serve to encourage further investigations into the history of the calculus.

Preface

vi

The inspiration toward the projection and completion of the present study has been due to Professor Barry, who has generously assisted in its prosecution through advice based on his wide familiarity with the field of the history of science. Professor Lynn Thorndike, of Columbia University, very kindly read and offered his competent criticism of the chapter on "Medieval Contributions." Professors L. P. Siceloff, of Columbia University, L. C. Karpinski, of the University of Michigan, and H. F. MacNeish, of Brooklyn College, have also read the manuscript and have furnished valuable aid and suggestions. Mrs. Boyer has been unstinting in her encouragement and promotion of the work, and has painstakingly done all of the typing. The composition of the Index has been undertaken by the Columbia University Press. Finally, from the American Council of Learned Societies came the subvention, in the form of a grant in aid of publication, which has made possible the appearance of this book in its present form. To all who have thus contributed toward the preparation and publication of this volume I wish to express my sincere appreciation. CARL B . BROOKLYN COLLEGE

January 3, 1939

BOYER

Contents I. INTRODUCTION II. CONCEPTIONS IN ANTIQUITY

1 14

III. MEDIEVAL CONTRIBUTIONS

61

IV. A CENTURY OF ANTICIPATION

96

V. NEWTON AND LEIBNIZ

187

VI. THE PERIOD OF INDECISION

224

VII. THE RIGOROUS FORMULATION

267

VIII. CONCLUSION

299

BIBLIOGRAPHY

311

INDEX

337

I. Introduction

M

ATHEMATICS has been an integral part of man's intellectual training and heritage for at least twenty-five hundred years. During this long period of time, however, no general agreement has been reached as to the nature of the subject, nor has any universally acceptable definition been given for it.1 From the observation of nature, the ancient Babylonians and Egyptians built up a body of mathematical knowledge which they used in making further observations. Thales perhaps introduced deductive methods; certainly the mathematics of the early Pythagoreans was deductive in character. The Pythagoreans and Plato2 noted that the conclusions they reached deductively agreed to a remarkable extent with the results of observation and inductive inference. Unable to account otherwise for this agreement, they were led to regard mathematics as the study of ultimate, eternal reality, immanent in nature and the universe, rather than as a branch of logic or a tool of science and technology. An understanding of mathematical principles, they decided, must precede any valid interpretation of experience. This view is reflected in the Pythagorean dictum that all is number,* and in the assertion attributed to Plato that God always plays the geometer.4 Later Greek skeptics, it is true, questioned the possibility of attaining any knowledge of such absolute character either by reason or by experience. But Aristotelian science had meanwhile shown that through observation and logic one can at least reach a consistent representation of phenomena, and mathematics consequently became, with Euclid, an idealized pattern of deductive relationships. Derived from postulates consistent with the results of induction from observation, it was found serviceable in the interpretation of nature. 1 Bell, The Queen of the Sciences, p. IS. For full citations of works referred to in the footnotes see the Bibliography. 1 See, for example, Republic VII, 527, in Dialogues, Jowett trans., Vol. II, pp. 362-63. »See Aristotle, Metaphysics 987a-989b, in Works, ed. by Ross and Smith, Vol. VIII; Cf. also 1090a. ' Plutarch, Miscellanies and Essays, HI, 402.

2

Introduction

The Scholastic view, which prevailed during the Middle Ages, was that the universe is "tidy" and simply intelligible. In the fourteenth century came a fairly clear realization that Peripatetic qualitative views of motion and variation could better be replaced by quantitative study. These two concepts, with a revival of interest in Platonic views, brought about in the fifteenth and sixteenth centuries a renewal of the conviction that mathematics is in some way independent of, and prior to, experiential and intuitive knowledge. Such conviction is marked in the thinking of Nicholas of Cusa, Kepler, and Galileo, and to a certain extent appears in that of Leonardo da Vinci. This conception of mathematics as the basis of the architecture of the universe was in turn modified in the sixteenth and seventeenth centuries. In mathematics, the cause of the change was the less critical and more practical use of the algebra which had been adopted from the Arabs, early in the thirteenth century, and then further developed in Italy. In natural science, the change was due to the rise of experimental method. The certitude in mathematics of which Descartes, Boyle, and others spoke was thus interpreted to mean a consistency to be found rather in the character of its reasoning than in any ontological necessity which it presented a priori. The centering of attention on the procedures rather than on the bases of mathematics was emphasized in the eighteenth century by an extraordinary success in applying the calculus to scientific and mathematical problems. A more critical attitude was inaugurated in the nineteenth century by persistent efforts to find a satisfactory foundation for the conceptions involved in this new analysis of the infinite. Mathematical rigor was revived, and it was discovered that Euclid's postulates are not categorical synthetic judgments, as Kant maintained, 4 but simply assumptions. Such premises, it was found, may be so freely and arbitrarily chosen that—subject to the condition that they be mutually compatible—they may be allowed to contradict the apparent evidence of the senses. Toward the close of the century, as the result of the arithmetizing tendency in mathematical analysis, it was further discovered that the concept of infinity, transcending all intuition and analysis, could be introduced into mathematics without impairing the logical consistency of the subject. » See Sttmrniliche Werke, II, passim.

Introduction

S

If the assumptions of mathematics are quite independent of the world of the senses, and if its elements transcend all experience,* the subject is at best reduced to bare formal logic and at worst to symbolical tautologies. The formal symbolic and arithmetizing tendency in mathematics has met with remarkable success in the study of the continuous. It has also led to stubborn paradoxes, a fact which has aroused increased interest in the nature of mathematics: its scope and place in intellectual life, the psychological source of its elements and postulates, the logical force of its propositions and their validity as interpretations of the world of sense perception. The old idea that mathematics is the science of quantity, or of space and number, has largely disappeared. The untutored intuition of space, it is realized, leads to contradictions, a fact which upsets the Kantian view of the postulates. Nevertheless the mathematician is guided, although he is not controlled, by the external world of sense perception.7 The mathematical theory of continuity originated in direct experience, but the definition of the continuum adopted in the end by the mathematician transcends sensory imagination. From this, mathematical formalists conclude that since we make no use of intuition in mathematical definitions and premises, it is not necessary that we should interpret the axioms or have any idea as to the nature of the objects and relations involved. The intuitionists, however, insist that the mathematical symbols involved should significantly express thoughts. 8 Although there are two (or more) views of the grounds for believing in the unassailable exactness of mathematical laws, the recognition that mathematical concepts are suggested, although not defined, by intuition thus easily accounts for the fact that the results of mathematical deductive reasoning are in apparent agreement with those of inductive experience. The derivative and the integral had their sources in two of the most obvious aspects of nature—multiplicity and variability—but were in the end defined as mathematical abstractions based on the fundamental concept of the limit of an • Bertrand Russell has taken advantage of this disconcerting situation to define mathematics facetiously as "the subject in which we never know what we are talking about, nor whether what we are saying is true." "Recent Work on the Principles of Mathematics," p. 84. ' B6cher, "The Fundamental Concepts and Methods of Mathematics." • Brouwer, "Intuitionism and Formalism."

4

Introduction

infinite sequence of elements. Once we have traced this development, the power and fecundity of these ideas when applied to the interpretation of nature will be easily understood. The calculus had its origin in the logical difficulties encountered by the ancient Greek mathematicians in their attempt to express their intuitive ideas on the ratios or proportionalities of lines, which they vaguely recognized as continuous, in terms of numbers, which they regarded as discrete. It became involved almost immediately with the logically unsatisfactory (but intuitively attractive) concept of the infinitesimal. Greek rigor of thought, however, excluded the infinitely small from geometrical demonstrations and substituted the circumventive but cumbersome method of exhaustion. Problems of variation were not attacked quantitatively by Greek scientists. No method could be developed which would do for kinematics what the method of exhaustion had done for geometry—indicate an escape from the difficulties illustrated by the paradoxes of Zeno. The quantitative study of variability, however, was undertaken in the fourteenth century by the Scholastic philosophers. Their approach was largely dialectical, but they had resort as well to graphical demonstration. This method of study made possible in the seventeenth century the introduction of analytic geometry and the systematic representation of variable quantities. The application of this new type of analysis, together with the free use of the suggestive infinitesimal and the more extensive application of numerical concepts, led within a short time to the algorithms of Newton and Leibniz, which constitute the calculus. Even at this stage, however, there was no clear conception of the logical basis of the subject. The eighteenth century strove to find such a basis, and although it met with little success in this respect, it did in the effort largely free the calculus from intuitions of continuous motion and geometrical magnitude. Early in the following century the concept of the derivative was made fundamental, and with the rigorous definitions of number and of the continuum laid down in the latter half of the century, a sound foundation was completed. Some twenty-five hundred years of effort to explain a vague instinctive feeling for continuity culminated thus in precise concepts which are logically defined but which represent extrapolations beyond the world of sensory ex-

Introduction

5

perience. Intuition, or the putative immediate cognition of an element of experience which ostensibly fails of adequate expression, in the end gave way, as the result of reflective investigation, to those well-defined abstract mental constructs which science and mathematics have found so valuable as aids to the economy of thought. The fundamental definitions of the calculus, those of the derivative and the integral, are now so clearly stated in textbooks on the subject, and the operations involving them are so readily mastered, that it is easy to forget the difficulty with which these basic concepts have been developed. Frequently a clear and adequate understanding of the fundamental notions underlying a branch of knowledge has been achieved comparatively late in its development. This has never been more aptly demonstrated than in the rise of the calculus. The precision of statement and the facility of application which the rules of the calculus early afforded were in a measure responsible for the fact that mathematicians were insensible to the delicate subtleties required in the logical development of the discipline. They sought to establish the calculus in terms of the conceptions found in the traditional geometry and algebra which had been developed from spatial intuition. During the eighteenth century, however, the inherent difficulty of formulating the underlying concepts became increasingly evident, and it then became customary to speak of the "metaphysics of the calculus," thus implying the inadequacy of mathematics to give a satisfactory exposition of the bases. With the clarification of the basic notions—which, in the nineteenth century, was given in terms of precise mathematical terminology—a safe course was steered between the intuition of the concrete in nature (which may lurk in geometry and algebra) and the mysticism of imaginative speculation (which may thrive on transcendental metaphysics). The derivative has throughout its development been thus precariously situated between the scientific phenomenon of velocity and the philosophical noumenon of motion. The history of the integral is similar. On the one hand, it has offered ample opportunity for interpretations by positivistic thought in terms either of approximations or of the compensation of errors-—views based on the admitted approximative nature of scientific measurements and on the accepted doctrine of superimposed effects. On the other hand, it has at the same time been regarded by idealistic

6

Introduction

metaphysics as a manifestation that beyond the finitism of sensory percipiency there is a transcendent infinite which can be but asymptotically approached by human experience and reason. Only the precision of their mathematical definition—the work of the past century— enables the derivative and the integral to maintain their autonomous position as abstract concepts, perhaps derived from, but nevertheless independent of, both physical description and metaphysical explanation. At this point it may not be undesirable to discuss these ideas, with reference both to the intuitions and speculations from which they were derived and to their final rigorous formulation. This may serve to bring vividly to mind the precise character of the contemporary conceptions of the derivative and the integral, and thus to make unambiguously clear the terminus ad quern of the whole development. The derivative is the mathematical device used to represent point properties of a curve or function. It thus has as its analogues in science instantaneous properties of a body in motion, such as the velocity of the object at any given time. When science is concerned with a time interval, the average velocity over this interval is suitably defined as the ratio of the change in the distance covered during the interval to the time interval itself. This ratio is conveniently represented by As the notation — . Inasmuch as the laws of science are formulated by At induction on the basis of the evidence of the senses, on the face of it there can be no such thing in science as an instantaneous velocity, that is, one in which the distance and time intervals are zero. The senses are unable to perceive, and science is consequently unable to measure, any but actual changes in position and time. The power of every sense organ is limited by a minimum of possible perception.® We cannot, therefore, speak of motion or velocity, in the sense of a scientific observation, when either the distance or the corresponding time interval becomes so small that the minimum of sensation involved in its measurement is not excited—much less when the interval is assumed to be zero. If, on the other hand, the distance covered is regarded as a function of the time elapsed, and if this relationship is represented mathe9

Cf. Mach, Die principien der Wärmelehre, pp. 71-77.

Introduction

7

matically by the equation s = f(t), the minimum of sensation no longer operates against the consideration of the abstract difference As quotient — . This has a mathematical meaning no matter how small At the time and distance intervals may be, provided, of course, that the time interval is not zero. Mathematics knows no minimum interval of continuous magnitudes—and distance and time may be considered as such, inasmuch as there is no evidence which would lead one to regard them otherwise. Attempts to supply a logical definition of such an infinitesimal minimum which shall be consistent with the body of mathematics as a whole have failed. Nevertheless, the term "instantaneous velocity" appears to imply that the time interval is to be regarded not only as arbitrarily small but as actually zero. Thus the term predicates the very case which mathematics is compelled to exclude because of the impossibility of division by zero. This difficulty has been resolved by the introduction of the derivative, a concept based on the idea of the limit. In considering the successive values of the difference quotient —, mathematics may conAt tinue indefinitely to make the intervals as small as it pleases. In this way an infinite sequence of values, n, rt, rs, . . . , rH, . . . (the successive values of the ratio

is obtained. This sequence may be such

that the smaller the intervals, the nearer the ratio rn will approach to some fixed value L, and such that by taking the value of n to be sufficiently large, the difference |L— r j can be made arbitrarily small. If this be the case, this value L is said to be the limit of the infinite sequence, or the derivative f'(t) of the distance function /(/), or the instantaneous velocity of the body. It is to be borne in mind, however, that this is not a velocity in the ordinary sense and has no counterpart in the world of nature, in which there can be no motion without a change of position. The instantaneous velocity as thus defined is not the division of a time interval into a distance interval, howsoever much the conventional notation — = /'(/) may dt suggest a ratio. This symbolism, although remarkably serviceable in

8

Introduction

the carrying out of the operations of the calculus, will be found to have resulted from misapprehension on the part of Leibniz as to the logical basis of the calculus. The derivative is thus defined not in terms of the ordinary processes of algebra, but by an extension of these to include the concept of the limit of an infinite sequence. Although science may not extrapolate beyond experience in thus making the intervals indefinitely small, and although such a process may be "inadequately adapted to nature," 10 mathematics is at liberty to introduce the new limit concept, on the basis of the logical definition given above. One can, of course, make this notion still more precise by eliminating the words "approach," "sufficiently large," and "arbitrarily small," as follows: L is said to be the limit of the above sequence if, given any positive number E (howsoever small), a positive integer N can be found such that for n > N the inequality \L — rn\ < e is satisfied. In this definition no attempt is made to determine any so-called "end" of the infinite sequence, or to deal with the possibility that the variable rn may "reach" its limit L. The number L, thus abstractly defined as the derivative, is not to be regarded as an "ultimate ratio," nor may it be invoked as a means of "visualizing" an instantaneous velocity or of explaining in a scientific or a metaphysical sense either motion or the generation of continuous magnitudes. It is such unclear considerations and unwarranted interpretations which, as we shall see, have embroiled mathematicians, since the time of Zeno and his paradoxes, in controversies which often misdirected their energy. On the other hand, however, it is precisely such suggestive notions which stimulated the investigations resulting in the formal elaboration of the calculus, even though this very elaboration was in the end to exclude them as logically irrelevant. Just as the problem of defining instantaneous velocities in terms of the approximation of average velocities was to lead to the definition of the derivative, so that of defining lengths, areas, and volumes of curvilinear configurations was to eventuate in the formulation of the definite integral. This concept, however, was likewise ultimately to be so defined that the geometrical intuition which gave it birth was excluded. As part of the Greek pursuit of unity in multiplicity, we 10

Schiodinger, Science and the Human Temperament, pp. 61-62.

9

Introduction

shall see that an attempt was made to inscribe successively within a circle polygons of a greater and greater number of sides in the hope of finally "exhausting" the area of the circle, that is, of securing a polygon with so great a number of sides that its area would be equal to that of the circle. This naive attempt was of course doomed to failure. The same process, however, was adopted by mathematicians as basic in the definition of the area of the circle as the limit A of the infinite sequence formed by the areas Au At, At, . . . , Aa, . . . of the approximating polygons. This affords another example of extrapolating beyond sensory intuition, inasmuch as there is no process by which the transition from the sequence of polygonal areas to the limiting area of the circle can be "visualized." An infinite subdivision is of course excluded from the realm of sensory experience by the fact that there exist thresholds of sensation. It must be banished also from the sphere of thought, in the physiological sense, inasmuch as psychology has shown that for an act of thought a measurable minimum of duration of time is required.11 Logical definition alone remains a sufficient criterion for the validity of this limiting value A. In order to free the limiting process just described from the geometrical intuition inherent in the notion of area, mathematics was constrained to give formal definition to a concept which should not refer to the sense experience from which it had arisen. (This followed a long period of indecision, the course of which we are shortly to trace.) After the introduction of analytical geometry it became customary, in order to find the area of a curvilinear figure, to substitute for the series of approximating polygons a sequence of sums of approximating rectangles, as illustrated in the diagram (fig. 1). The area of each of the rectangles could be represented by the notation /(jr^Ai,»

and the sum of these by the symbolism Sa = 2 f(xf)Ax{. The area of the figure could then be defined as the limit of the infinite sequence of sums Sm as the number of subdivisions n increased indefinitely and as the intervals Aar, approached zero. Having set up the area in this manner with the help of the analytical representation of the curve, it then became a simple matter to discard the geometrical intuition leading to the formation of these sums and to define the definite u

Enriques, Problems of Science, p. IS.

10

Introduction

integral of j{x) over the interval from x = a to x = b arithmetically as the limit of the infinite sequence of sums S„ =

»

Z f(x{) A*,- (where

the divisions A x t are taken to cover the interval from a to b) as the intervals A x t become indefinitely small. This definition then invokes, apart from the ordinary operations of arithmetic, only the concept of the limit of an infinite sequence of terms, precisely as does that of the derivative. The realization of this fact, however, followed only after many centuries of investigation by mathematicians. The very notation i"j(x)dx, which is now customarily employed to represent this definite integral, is again the result of the historical development of

formulation. It is suggestive of a sum, rather than of the limit of an infinite sequence; and in this respect it is in better accord with the views of Leibniz than with the intention of the modern definition. The definite integral is thus defined independently of the derivative, but Newton and Leibniz discovered the remarkable property constituting what is commonly known as the fundamental theorem of the calculus, viz., that the definite integral F(x) = flf(x)dx of the continuous function f(x) has a derivative which is this very same function, F'(x) = /(x). That is, the value of the definite integral of f(x) from a to b may in general be found from the values, for x = a and x = b, of the function F(x) of which J(x) is the derivative. This relationship between the derivative and the definite integral has been called "the root idea of the whole of the differential and integral

11

Introduction 11

calculus." The function F(x), when so defined, is often called the indefinite integral of /(x), but it is to be recognized that it is in this case not a numerical limit given by an infinite sequence, as is the definite integral, but is a function of which /(*) is the derivative. The function F(x) is sometimes also called the primitive of /(*), and the value of F(b) — F(a) is occasionally taken as the definition of yaf(x)dx.

.

In this case the relationship j,f(x)dx

= lirn H-MO

»

z J(xi)Ax{ i-1

is

then the fundamental theorem of the calculus, rather than the definition of the definite integral. Although the recognition of this striking inverse relationship, together with the formulation of rules of procedure, may be taken as constituting the invention of the subject, it is not to be supposed that the inventors of the calculus were in possession of the above sophisticated concepts of the derivative and the integral, so necessary in the logical development of the new analysis. More than a hundred years of investigation was to be required before the achievement of their final definition in the nineteenth century. It is the purpose of this essay to trace the development of these two concepts from their incipiency in sense experience to their final elaboration as mathematical abstractions, defined in terms of formal logic by means of the idea of the limit of an infinite sequence. We shall find that the history of the calculus affords an unusually striking example of the slow formation of mathematical concepts by the emancipation from all sense data of ideas born of our primary intuitions. The derivative and the integral are, in the last analysis, synthetically defined in terms of ordinal considerations and not of those of continuous quantity and variability. They are, nevertheless, the results of attempts to schematize our sense impressions of these last notions. This explains why the calculus, in the early stages of its development, was bound up with concepts of geometry or motion, and with explanations of indivisibles and the infinitely small; for these ideas are suggested by naive intuition and experience of continuity. There is in the calculus a further concept which merits brief consideration at this point, not so much on account of any logical exigencies in the present structure of the calculus as to make clearer its historical development. The infinite sequences considered above in u

Couiant, Dijfereniial and Integral Calculus, II, 111.

12

Introduction

the definitions of the derivative and of the integral were obtained by continuing, in thought, to diminish ad infinitum the intervals between the values of the independent variable. Considerations which in physical science led to the atomic theory were at various periods in the development of the calculus adduced in mathematics. These made it appear probable that just as in the actual subdivision of matter (which has the appearance of being continuous) we arrive at ultimate particles or atoms, so also in continuous mathematical magnitudes we may expect (by means of successive subdivisions carried on in thought) to obtain the smallest possible intervals or differentials. The derivative would in this case be defined as the quotient of two such differentials, and the integral would then be the sum of a number (perhaps finite, perhaps infinite) of such differentials. There is, to be sure, nothing intuitively unreasonable in such a view; but the criterion of mathematical acceptability is logical selfconsistency, rather than reasonableness of conception. Such a view of the nature of the differential, although possessing heuristic value in the application of the calculus to problems in science, has been judged inacceptable in mathematics because no satisfactory definition has as yet been framed which is consistent with the principles of the calculus as formulated above, or which may be made the basis of a logically satisfactory alternative exposition. In order to retain the operational facility which the differential point of view affords, the concept of the differential has been logically defined, not in terms of mathematical atomism, but as a notion derived from that of the derivative. The differential dx of an independent variable x is to be thought of as nothing but another independent variable; but the differential dy of a function y = f(x) is defined as that variable the values of which are so determined that for any given value of the variable dx the dy ratio — shall be equal to the value of the derivative at the point in dx question, i. e., dy = f'(x)dx. The differentials as thus defined are only new variables, and not fixed infinitesimals, or indivisibles, or "ultimate differences," or "quantities smaller than any given quantity," or "qualitative ieros,"13 or "ghosts of departed quantities," as they " "Abominable little zeroes," they have been called by Osgood. See Osgood, "The Calculus in Our Colleges and Technical Schools"; and Huntington, "Modem Interpretation of Differentials."

Introduction

13

have been variously considered in the development of the calculus. Poincaré has said that had mathematicians been left the prey of abstract logic, they would never have gotten beyond the theory of numbers and the postulates of geometry." It was nature which thrust upon mathematicians the problems of the continuum and of the calculus. It is therefore quite understandable that the persistent atomistic speculations of physical thought should have had a counterpart in attempts to picture, by means of indivisible elements, the space described by geometry. The further development of mathematics, however, has shown that such notions must be abandoned, in order to preserve the logical consistency of the subject. The basis of the concepts leading to the derivative and the integral was first found in geometry, for despite the apodictic character of its proofs, this subject was considered an abstract idealization of the world of the senses. Recently, however, it has been more clearly perceived that mathematics is the study of relationships in general and must not be hampered by any preconceived notions, derived from sensory perception, of what these relationships should be. The calculus has therefore been gradually emancipated from geometry and has been made dependent, through the definitions of the derivative and the integral, on the notion of the natural numbers, an idea from which all traditional pure mathematics, including geometry, can be derived.16 Mathematicians now feel that the theory of aggregates has provided the requisite foundations for the calculus, for which men had sought since the time of Newton and Leibniz.16 It is impossible to predict with any confidence, however, that this is the final step in the process of abstracting from the primitive ideas of change and multiplicity all those irrelevant incumbrances with which intuition binds these concepts. It is a natural tendency of man to hypostatize those ideas which have great value for him,17 but a just appreciation of the origin of the derivative and the integral will make clear how unwarrantedly sanguine is any view which would regard the establishment of these notions as bringing to its ultimate close the development of the concepts of the calculus. 14

See Osgood, op. cil., p. 457; Poincaré, Foundations of Science, p. 46. Russell, Introduction to Mathematical Philosophy, p. 4. See also Poincaré, Foundations of Science, pp. 441, 462. w Russell, The Principles of Mathematics, pp. 325-26. 17 Mach, The Science of Mechanics, p. 541. u

II. Conceptions in Antiquity

T

HE PRE-HELLENIC peoples are usually regarded as prescientific in their attitude toward nature,1 inasmuch as they palpably lacked the Greek confidence in its essential reasonableness, as well as the associated feeling that beneath the perplexing heterogeneity and ceaseless flux of events would be found elements of uniformity and permanence. The search for universals, which the Greeks maintained so persistently, apparently held no attraction for the Egyptians and the Babylonians. So also the mathematical thought of these peoples— about whom, among those of the ancient world, we are best informed— bore no significant resemblance to ours, in that it lacked the tendency, essential to both mathematical and scientific method, toward the isolation and abstraction of certain samenesses from their confusingly varied concomitants in nature and in thought. Lacking these elements of invariance to serve as premises of inference, they were accordingly without appreciation of the characteristics which distinguish mathematics from science, namely, its logical nature and the necessity of deductive proofs.2 A large body of knowledge of spatial and numerical relations they did, however, acquire; and the more familiar their work becomes, the more it inspires our admiration.® It was, however, largely the result of empirical investigations, or at best of generalizations which were the result of incomplete induction from simple to more complicated cases. The Egyptian rule for computing the volume of a square pyramid—from which was obtained the most remarkable of all 1 On this point, however, there are significant differences of opinion. Barry (The Scientific Habit of Thought, p. 104) places "the childhood of science" among the early Greeks; Bumet (Greek Philosophy, Part I, "Thales to Plato," pp. 4-5) says "natural science is the creation of the Greeks," and finds "not the slightest trace of that science in Egypt or even in Babylon." On the other hand, Karpinski ("Is There Progress in Mathematical Discovery," pp. 51—52) would regard the achievements of the Babylonians, Egyptians, and Hindus as "scientific in the highest sense." ' Milhaud, NouveUes itudes sur I'histoire de la pensie scientifique, pp. 41-133. ' See Neugebauer, Vorlesungen Uber Gesckichte der aniiken mathematischeit Wissenschafien, Vol. I, Vorgriechische Mathematik, for the best account of this work.

Conceptions in Antiquity

15

Egyptian results, the rule for determining the volume of a frustum of a square pyramid—was probably the result of this method of procedure.4 That the demonstration could not be correct, in our understanding of the mathematical implications of this term, is clear from the fact that this result for the general case requires the use of infinitesimal or limit considerations' which, while constituting the point of departure for the story of the derivative and integral, are not found in any record before the Greek period.6 More fundamental than this lack of deductive proofs of inferred results is the fact that in all this Egyptian work the rules were applied to concrete cases with definite numbers only.7 There was no conception in their geometry of a triangle as representative of all triangles,8 an abstract generalization necessary for the elaboration of a deductive system. This lack of freedom and imagination is apparent also in Egyptian arithmetic, into which the abstract number concept, as such, did not enter,* and in which, with the exception of i , all rational fractions were expressed as sums of unit fractions.10 Babylonian mathematics resembles the Egyptian more than the Greek, but with a stronger emphasis on the numerical side and thus a more highly developed algebra than that of Egypt. Here again we must not look for logical structure or proof, more complicated cases being reduced to simpler, and so "proved," or rather treated as analogous without proof;11 and we must remember that this work, like the Egyptian, deals with concrete cases only. In connection with Babylonian astronomy, we find that problems involving continuous variation were studied, but only to the extent of tabulating the values of a function (such as the brightness of the moon, for example) for * Ibid., p. 128. ' See Dehn, "Ueber raumgleiche Polyeder," for proof that infinitesimal considerations cannot here be avoided. • Neugebauer, op. cit., pp. 126-28. There seems to be no basis for the implication made by Bell (The Search for Truth, p. 191) that the Egyptians used infinite or infinitesimal considerations in deductive reasoning. ' Neugebauer, op. cit., p. 127. • Luckey, "Was ist ägyptische Geometrie?" p. 49. »Neugebauer, op. cit., p. 203. See also Miller, "Mathematical Weakness of the Early Civilizations." 10 For example, the Egyptians did not regard as a single number that number which we now represent by the symbol g, but thought of it as the sum of the two fractions, £ and ^ . u Neugebauer, op. cit., pp. 203-4.

16

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values of the argument (time) measured at equal intervals, and from this calculating the maximum (intensity) of the function.15 The Greeks were the first, however, systematically to analyze1® the idea of continuous magnitude and to develop concepts leading to the integral and the derivative. Our information on the history of mathematics in the interval between that of the best Egyptian and Babylonian mathematics and the early work in Greece is unfortunately fragmentary. That these oriental civilizations influenced Greek culture is clear; but the nature and extent of their contribution is undetermined. However that may be, there is an obvious change in spirit in both science and mathematics, as these developed in Greece. The human mind was "discovered" as something different from the surrounding body of nature and capable of discerning similarities in a multiplicity of events, of abstracting these from their settings, generalizing them, and deducing therefrom other relationships consistent with further experience. It is for this reason that we consider mathematical and scientific method as originating with the Hellenic race;14 but to say that Greek mathematics and science were autochthonous would be to forget the debt of subject-matter owed to Egypt and Babylon.16 It is likely" that the new outlook of the Hellenes was the result of the flux of civilizations occurring at this time, this impressing upon the rising Greek fortunes the stamp of numerous cultures. Thales is the first Greek mentioned in connection with this "intellectual revolution," which produced elementary mathematics and which was to reveal those difficulties in conception, the study and resolution of which were to produce within the next twenty-five hundred years the subject which we now call the calculus. He is said to have been a great traveler, to have learned geometry from the Egyptians and astronomy from the Babylonians, and upon his return to Greece to have instructed his successors in the principles of these Hoppe, "Zur Geschichte der Infinitesimaliechnung bis Leibniz und Newton." Neugebauer, op. cit., p. 20S. " T. L. Heath, A History of Greek Mathematics, I, v, says, "Mathematics in short is a Greek science, whatever new developments modern analysis has brought or may bring." u See Karpinski, "Is There Progress in Mathematical Discovery?" pp. 46-47. Cf. also Gandz, "The Origin and Development of the Quadratic Equations in Babylonian, Greek and Early Arabic Algebra," pp. 542-43. w As Neugebauer suggests (op. cit., p. 203). u u

Conceptions in Antiquity

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subjects. Proclus says of his method of attack that it was "in some cases more general, in other cases more empirical." Thaïes' demonstrations may therefore have appealed to some extent to the evidence of the senses, and in fact his theorems were those the truth of which one would recognize by the execution of some practical construction." To Thaïes, nevertheless, is ascribed the establishment of mathematics as a deductive discipline.18 He did not, however, construct a body of mathematical knowledge, nor did he apply his method to the analysis of the problem of the continuum. These tasks seem to have been performed by Pythagoras, the second Greek mathematician of whom we have substantial information. According to Proclus, he "transformed the study of geometry into a liberal education, examining the principles of the science from the beginning and probing the theorems in an immaterial and intellectual manner"; 1 ' but, beyond admitting this, it is impossible to ascribe with any degree of certainty other mathematical or scientific accomplishments to Pythagoras the individual, since never in antiquity could he be distinguished from his school and it is hardly possible to do so now. The knowledge acquired by the school established by Pythagoras was held to be strictly esoteric, with the result that when the general nature of Pythagorean thought became apparent, after the death of the founder about 500 B. c., it was already impossible to attribute to a single member any one contribution. Nevertheless, the process of abstraction begun by Thaïes was evidently completed by this school. A new difficulty, however, then entered into Greek thought, for the Pythagorean mathematical concepts, abstracted from sense impressions of nature, were now in turn projected into nature and considered to be the structural elements of the universe.20 Thus the Pythagoreans attempted to construct the whole heaven out of numbers, the stars being units which were material points. Later they identified the regular geometrical solids, with which they were familiar, with the different sorts of substances in nature.21 Geometry was regarded 17 Paul Tannery, La Gtomitrie grecque, pp. 89 ff. « T. L. Heath, History of Greek Mathematics, I, 128. " Ibid., I, 140. Cf. also Moritz Cantor, Vorlesimgen fiber Geschichte der Mathematik, I, 137. " See Brunschvicg, Les Étapes de la philosophie mathématique, pp. 34 ff. n T. L. Heath, History of Greek Mathematics, I, 165.

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by them as immanent in nature, and the idealized concepts of geometry appeared to them to be realized in the material world. This confusion of the abstract and the concrete, of rational conception and empirical description, which was characteristic of the whole Pythagorean school and of much later thought, will be found to bear significantly on the development of the concepts of the calculus. It has often been inexactly described as mysticism,22 but such stigmatization appears to be somewhat unfair. Pythagorean deduction a priori having met with remarkable success in its field, an attempt (unwarranted, it is now recognized) was made to apply it to the description of the world of events, in which Ionian hylozoistic interpretations a posteriori had made very little headway. This attack on the problem was highly rational and not entirely unsuccessful, even though it was an inversion of the scientific procedure, in that it made induction secondary to deduction. One very important result of the Pythagorean search for unity in nature and geometry was the theory of application of areas. This originated with the Pythagoreans, if not with Pythagoras himself,23 and became fundamental in Greek geometry, in which it later led to the method of exhaustion, the Greek equivalent of our integration.24 The method of the application of areas enabled them to say of a figure bounded by straight lines that it was greater than, equivalent to, or less than2® a second figure. Such a superposition of one area upon another constitutes the first step in the attempt to make exactly definable the notion of area, in which a unit of area is said to be contained in a second area a given number of times. Modern mathematics has made fundamental the concept of number rather than that of congruence, with the result that the word "area" no longer calls vividly to mind that comparison of two surfaces which is essential in this connection and which was always uppermost in Greek thought. Greek mathematicians did not speak of the area of one figure, but of the ratio of two surfaces, a definition which could not, because of the problem of incommensurability, be made precise before a satisfactory concept of number had been developed. Such a concept the Pythag° See Russell, Our Knowledge of the External World, p. 19. T. L. Heath, History of Greek Mathematics, I, 150. u It was basic also in the Greek solution of quadratic equations by geometrical algebra. u Our names for the conic sections (ellipse, hyperbola, and parabola) were, incidentally, derived from the designations the Pythagoreans used in this connection. a

Conceptions in Antiquity

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oreans did not possess. This contribution was not made until the last half of the nineteenth century, and it was to furnish, in the last analysis, the basis of the whole of the calculus. However, to the Pythagoreans, in all probability, we owe the recognition of the need for some such concept-—a discovery which may be regarded as the first step, a terminus a quo, in the development of the concepts of the calculus. The inadequacy of the Pythagorean view of the ratio of magnitudes was first made evident to the followers of this school on the application of the doctrine, not to areas, but rather to the analogous comparison of lines which is presupposed by our notion of length. Such investigations led the Pythagoreans to an intensely disconcerting discovery. If the side of a square were to be applied to the diagonal, no common measure could be discovered which would express one in terms of the other. In other words, these lines were shown to be incommensurable. Just when this discovery took place and whether it was made by Pythagoras himself, by the early Pythagoreans, or by later members of the school are moot points in the history of mathematics. 2 ' It has also been maintained that Pythagoras owed his knowledge of the irrational and of the five regular solids, as well as much of his philosophy, to the Hindus.27 The question as to how the incommensurability was discovered or proved is also difficult to answer with any assurance. The method of application would suggest as a form of proof the geometrical equivalent of the process of finding the greatest common divisor, but there is another aspect of Pythagorean thought which points to a different sort of reasoning. A prevailing belief in the unity and harmony of nature and knowledge had led the Pythagoreans not only to explain different aspects of nature by various mathematical abstractions, as already suggested, but also to attempt to identify the realms of number and magnitude.28 By the term number, however, the Pythagoreans did " On this question see the following two papers by Heinrich Vogt: "Die Entdeckungsgeschichte des Irrationalen nach Plato und anderen Quellen des 4. Jahrhunderts": "Zur Entdeckungsgeschichte des Irrationalen." Vogt concludes that the discovery was made by the later Pythagoreans at some time before 410 B. C. T. L. Heath (History of Greek Mathematics, I, 157) would place it "at a date appreciably earlier than that of Democritus." " See Schroeder, Pythagoras und die Inder; and Vogt, "Haben die alten Inder den Pythagoreischen Lehrsatz und das Irrationale gekannt?" " For a keenly critical account of the significance, from the scientific point of view, of the Pythagorean problem of associating the fields of number and magnitude, see Barry, The Scientific Habit of Thought, pp. 207 ff.

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not understand the abstraction to which we give this name, but used it to designate " a progression of multitude beginning from a unit and a regression ending in i t . " " The integers were thus fundamental, numbers being collections of units, and, as was the case with their geometrical forms, they were immanent in nature, each having a position and occupying a place in space. If geometrical abstractions were the elements of actual things, number was the ultimate element of these abstractions and thus of physical bodies and of all nature.*0 This hypostatization of number had led the Pythagoreans to regard a line as made up of an integral number of units. This doctrine could not be applied to the diagonal of a square, however, for no matter how small a unit was chosen as a measure of the sides, the diagonal could not be a "progression of multitude" beginning with this unit. The proof of this fact, as given by Aristotle (and which possibly is that of the Pythagoreans),' 1 is based on the distinction between the odd and the even, which the Pythagoreans themselves had emphasized. The incommensurability of lines remained ever a stumbling block for Greek geometry. That it made a strong impression on Greek thought is indicated by the story, repeated by Proclus, that the Pythagorean who disclosed the fact of incommensurability suffered death by shipwreck as a result. It is demonstrated also, and more reliably, by the prominence given to the doctrine of irrationals by Plato and Euclid. It never occurred to the Greeks to invent an irrational number32 to circumvent the difficulty, although they did develop as a part of geometry (found, for example, in the tenth book of Euclid's Elements) the theory of irrational magnitudes. Failing to generalize their number system along the lines suggested later by the development of mathematical analysis, the only escape for Greek mathematicians in the end was to abandon the Pythagorean attempt to identify the realm of number with that of geometry or of continuous magnitude. The effort to unite the two fields was not given up, however, before intuition had sought another way out of the difficulty. If there is no » T. L. Heath, History of Greek Mathematics, I, 69-70. Milhaud, Les Philosophes géomètres de la Grèce, p. 109. " Zeuthen, "Sur l'origine historique de la connaissance des quantités irrationelles." n Stolz, Vorlesungen iiber AUgemeine Arithmetik, 1,94; cf. also Vogt, Der Grentbegriff in der Elementar-mothematik, p. 48. 10

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finite line segment so small that the diagonal and the side may both be expressed in terms of it, may there not be a monad or unit of such a nature that an indefinite number of them will be required for the diagonal and for the side of the square? We do not know definitely whether or not the Pythagoreans themselves invoked the infinitely small. We do know, however, that the concept of the infinitesimal had entered into mathematical thought, through a doctrine elaborated in the fifth century B. C., as the result of Greek speculation concerning the nature of the physical world. After the failure of the early Ionian attempts to find a fundamental element out of which to construct all things, there arose at Abdera the materialistic doctrine of physical atomism, according to which there is no one physis, not even a small group of substances of which everything is composed. The Abderitic school held that everything, even mind and soul, is made up of atoms moving about in the void, these atoms being hard indivisible particles, qualitatively alike but of countless shapes and sizes, all too small to be perceived by sense impressions. There is nothing either logically or physically inconsistent in this doctrine, which is a crude anticipation of our own chemical thought; but the greatest of the Greek atomists, Democritus, did not stop here: he was also a mathematician and carried the idea over into geometry. As we now know from the Method of Archimedes, which was discovered as a palimpsest in 1906, Democritus was the first Greek mathematician to determine the volumes of the pyramid and the cone. How he derived these results we do not know. The formula for the volume of a square pyramid was probably known to the Egyptians,53 and Democritus in his travels may have learned of it and generalized the result to include all polygonal pyramids. The result for the cone would then be a natural inference from the result of increasing indefinitely the number of sides in a regular polygon forming the base of a pyramid. This explanation would correspond to others involving similar infinitesimal conceptions, which we know Democritus entertained and which later influenced Plato.34 a Neugebauer, Vorlesungen über Geschichte der Atüiken p. 128. u See Luria, "Die Infinitesimaltheorie der antiken Atomisten."

mathematischen Wissenschaften,

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Aristotle and Euclid ascribe to him a mathematical atomism, and we know from Plutarch** that he was puzzled as to whether the infinitesimal parallel circular sections of which the cone may be considered to be composed are equal or unequal: if they are equal, the cone would be equal to the circumscribed cylinder; but if unequal, they would be idented like steps.®4 We do not know how he resolved this aporia, but it has been suggested that he made use of the idea of infinitely thin circular laminae, or indivisibles, to find the volumes of cones and cylinders, anticipating and using Cavalieri's theorem for these special cases.*7 Democritus seems to have discriminated clearly between physical and mathematical atoms—as did his later follower Epicurus, although Aristotle made no such distinction" and according to the much later account of Simplicius Democritus is said to have held that all lines are divisible to i n f i n i t y H o w e v e r , since most of Democritus' work is lost, we cannot now reconstruct his thought. That he was interested in other mathematical problems bearing on the infinitesimal we know from the titles of works now lost, but which are referred to by Diogenes Laertius. One of these seems to have been on horn angles (the angles formed by curves which have a common tangent at a point), and another on irrational (incommensurable) lines and solids.40 I t may be inferred, therefore, that the Pythagorean difficulty with the incommensurable was probably familiar to him, and it may be that he tried to solve it by some theory of mathematical atomism. It has been maintained41 that Democritus was too good a mathematician to have had anything to do with such a theory as that of indivisible lines; but it is difficult to imagine how a mathematical atom is to be conceived if not as an indivisible. At all events, whatever his conception of the nature of infinitesimals may have been, the influence of Democritus has persisted. The idea of the fixed infinitesimal magnitude has clung tenaciously to mathematics, frequently to be invoked by intuition when logic apparently failed to offer a solution, and finally to be displaced in the last century by the rigorous concepts of the derivative and the integral. " Plutarch, Miscellanies and Essays, IV, 414-16. " See also T. L. Heath, History of Greek Mathematics, I, 180; and Luria, op. cit., pp. 138-40, for statements of this "paradox." " Simon, Geschichte der Mathematik im Altertum, p. 181. " Luria, op. cit., pp. 179-80. ** Cf. Simplicii commentarii in ocio Aristotelis physicae auscultationis libros, p. 7. a Ibid., p. 181. « T. L. Heath, History of Greek Mathematics, I, 179-81.

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That the infinitesimal was not eagerly welcomed into Greek geometry after the time of the Pythagoreans and Democritus may have been due largely to a school of philosophy that had risen at Elea, in Magna Graecia. The Eleatic school, although not essentially mathematical, was apparently familiar with, and probably influenced by, Pythagorean mathematical philosophy; but it became an opponent of the chief tenet of this thought. Instead of proclaiming the constitution of objects as an aggregate of units, it pointed out the apparent contradictions inherent in such a doctrine, maintaining against the atomic view the essential oneness and changelessness of the world. This stultifying monism was upheld by Parmenides, the leader of the school, with perhaps a touch of skepticism derived from his iconoclastic predecessor, the poet-philosopher Xenophanes. In an indirect defense of this doctrine, the Eleatics proceeded to demolish, with skillful dialectic, the basis of opposing schools of thought. The most damaging arguments were offered by Zeno, the student of Parmenides. After presenting the obvious objection to the Pythagorean indefinitely small monad—that if it has any length, an infinite number will constitute a line of infinite length; and if it has no length, then an infinite number will likewise have no length—he added the following general dictum against infinitesimals: "That which, being added to another does not make it greater, and being taken away from another does not make it less, is nothing."42 More critical and subtle than these, however, are his four famous paradoxes on motion.43 There has been much speculation as to the purpose of Zeno's arguments,44 lack of evidence making it impossible to decide conclusively against whom they were directed: whether against the Pythagoreans, or the atomists, or Heraclitus, or whether they were mere sophisms. That they were intended merely as dialectical puzzles may perhaps be indicated by the passage in Plutarch's life of Pericles: Also the two-edged tongue of mighty Zeno, who, Say what one would, could argue it untrue.46 41

Zeller, Die Pkilosophie der Griechen in ihrer Geschichilichen Enturicklung, I, 540. For an unusually extensive account of the history of Zeno's paradoxes, with bibliography, see Cajori's article on "History of Zeno's Arguments on Motion." " O n this subject see Cajori, "The Purpose of Zeno's Arguments on Motion." This article includes an account of the varying interpretations, as well as extensive bibliographical notes. " Plutarch, The Lives of the Noble Grecians and Romans, p. 185. a

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On the other hand there is some reason to suppose that these arguments were presented in connection with a more significant purpose. It is not improbable that Zeno, although he was neither a mathematician nor a physicist, propounded the paradoxes to point out the weakness in the Pythagorean definition of a point as unity having position, and in the resulting Pythagorean multiplicity which did not distinguish clearly between the geometrical and the physical.44 Pythagorean science and mathematics had been concerned with form and structure, and not with flux and variability; but had the Pythagoreans applied their philosophy to the aspects of change in nature rather than to those of permanence, the resulting explanation of motion would have been in terms of concepts attacked by Zeno in his third and fourth paradoxes (those of the arrow and the stade),47 in which space and time are assumed to be composed of indivisible elements. The arguments would hold equally well, of course, against mathematical atomism. The first two paradoxes (the dichotomy and the Achilles**) are directed against the opposite conception, that of the infinite divisibility of space and time, and are based upon the impossibility of conceiving intuitively the limit of the sum of an infinite series. The four paradoxes are, of course, easily answered in terms of w See Paul Tannery, La Giométrie grecque, p. 124; cf. also the same author, "Le Concept scientifique du continu. Zénon d'Elée et Georg Cantor." Milhaud (Nouvdles (ludes, pp. 153-54) and Cajori concur in the view here presented. See further, Cajori, "Purpose of Zeno's Arguments." c The argument in the arrow is as follows: Anything occupying space equal to itself (or in one and the same place) is at rest; but this is true of the arrow at every moment of its flight. Therefore the arrow does not move. (See The Works of Aristotle, Vol. II, Pkysica VI. 239b, for the statement of the paradoxes.) The argument in the stade, as given by Aristotle, is obscure (because of brevity), but is equivalent to the following: Space and time being assumed to be made up of points and instants, let there be given three parallel rows of points, A, B, and C. Let C move to the right and A to the left at the rate of one point per instant, both relative to B; but then each point of A will move past two points of C in an instant, so that we can subdivide this, the smallest interval of time; and this process can be continued ad infinitum, so that time can not be made up of instants. 48 The argument in the dichotomy is as follows: before an object can traverse a given distance, it must first traverse half of this distance; before it can cover half, however, it must cover one quarter; and so ad infinitum. Therefore, since the regression is infinity motion is impossible, inasmuch as the body would have to traverse an infinite number of divisions in a finite time. The argument in the A chilles is similar. Assume a tortoise to have started a given distance ahead of Achilles in a race. Then by the time Achilles has reached the starting point of the tortoise, the latter will have covered a certain distance; in the time required by Achilles to cover this additional distance, the tortoise will have gone a little farther; and so ad infinitum. Since this series of distances is infinite, Achilles can never overtake the tortoise, for the same reason as that adduced in the dichotomy.

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the concepts of the differential calculus. There is no logical difficulty in the dichotomy or the Achilles, the uneasiness being due merely to failure of the imagination to realize, in terms of sense impressions, the nature of infinite convergent series which are fundamental in the precise explanation of, but not involved in our obscure notion of, continuity. The paradox of the flying arrow involves directly the conception of the derivative and is answered immediately in terms of this. The argument in this paradox, as also that in the stade, is met by the assumption that the distance and time intervals contain an infinite number of subdivisions. Mathematical analysis has shown that the conception of an infinite class is not self-contradictory, and that the difficulties here, as also in the case of the first two paradoxes, are those of conceiving intuitively the nature of the continuum and of infinite aggregates. 49 In a broad sense there are no insoluble problems, but only those which, arising from a vague feeling, are not yet suitably expressed.50 This was the position of Zeno's paradoxes in Greek thought; for the notions involved were not given the precision of expression necessary for the resolution of the putative difficulties. It is clear that the answers to Zeno's paradoxes involve the notions of continuity, limits, and infinite aggregates—abstractions (all related to that of number) to which the Greeks had not risen and to which they were in fact destined never to rise, although we shall see Plato and Archimedes occasionally straining toward such views. That they did not do so may have been the result of their failure, indicated above in the case of the Pythagoreans, clearly to separate the worlds of sense and reason, of intuition and logic. Thus mathematics, instead of being the science of possible relations, was to them the study of situations thought to subsist in nature. The inability of Greek mathematicians to answer in a clear manner the paradoxes of Zeno made it necessary for them to forego the attempt to give to the phenomena of motion and variability a quantitative explanation. These experiences were consequently confined to the field of metaphysical speculation, as in the work of Heraclitus, or • Accounts of the mathematical resolutions of Zeno's paradoxes are given in the works of Bertrand Russell, The Principles of Mathematics and Our Knowledge of the External World. 60 Knriques, Problems of Science, p. 5.

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to that of qualitative description, as the physics of Aristotle. Only the static aspects of optics, mechanics, and astronomy found a place in Greek mathematics, and it remained for the Scholastics and early modern scientists to establish a quantitative dynamics. Zeno's arguments and the difficulty of incommensurability had also a more general effect on mathematics: in order to retain logical precision, it was necessary to give up the abortive Pythagorean effort to identify the domains of number and geometry, and to abandon also the premature Democritean attempt to explain the continuous in terms of the discrete. It is, however, impossible satisfactorily to interpret the world of nature and the realm of geometry (spheres which for the Greeks were not essentially distinct) without superimposing upon them a framework of discrete multiplicity; without ordering, by means of number, the heterogeneity of impressions received by the senses; and without at every point comparing nonidentical elements. Thought itself is possible only in terms of a plurality of elements. As a consequence, the concept of discreteness cannot be excluded completely from the study of geometry. The continuous is to be interpreted in terms of successive subdivision, that is to say, in terms of the discrete, although from the Greek point of view the former could not be logically identified with the latter. The clever manner in which the method of successive subdivision was applied in Greek geometry, without the loss of logical rigor, will be seen later in the method of exhaustion—a procedure which was developed, not in Italy, but in and around the Greek mainland, whither many Pythagoreans wandered, on the breaking up of the school, toward the beginning of the fifth century b. c. Zeno likewise lived for a time in Athens, the rising center of Greek culture and mathematics. Here Pericles, the political leader of that city in its Golden Age, is said to have been one of his listeners." At Athens the great philosopher Plato, although himself not primarily a mathematician, was conversant with, and displayed a lively interest in, the problems of the geometers. He may not have contributed much original work in mathematics, but he advanced the subject, nevertheless, through his great enthusiasm for it. He is said to have paid particular attention to the principles of geometry—to the 11

Plutarch, Lives, p. 185.

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87

1

hypotheses, definitions, methods.* For this reason he was particularly concerned with the difficulties which led eventually to the calculus. In his dialogues he considered the Pythagorean problem of the nature of number and its relationship to geometry," the difficulty of incommensurability,44 the paradoxes of Zeno,46 and the Democritean question of indivisibles and the nature of the continuum.4® Plato seems to have realized the gulf between arithmetic and geometry, and it has been conjectured47 that he may have tried to bridge it by his concept of number and by the establishment of arithmetic upon a firm axiomatic basis similar to that which was built up in the nineteenth century independently of geometry; but we cannot be sure, because these thoughts do not occur in his exoteric writings and were not advanced by his successors. If Plato made an attempt to arithmetize mathematics in this sense, he was the last of the ancients to do so, and the problem remained for modern mathematical analysis to solve. The thought of Aristotle we shall find diametrically opposed to any such conceptions. It has been suggested that Plato's thought was so opposed by the polemic of Aristotle that it was not even mentioned by Euclid. Certain it is that in Euclid there is no indication of such a view of the relation of arithmetic to geometry; but the evidence is insufficient to warrant the assertion4* that, in this connection, it was the authority of Aristotle which held back for two thousand years a transformation which the Academy sought to complete. A sound basis for either mechanics or arithmetic must be built upon the limit concept—a notion which is not found in the extant works of Plato nor in those of his successors. The Platonists, on the contrary, attempted to develop the misleading idea of indivisibles or fixed infinitesimals, a notion which the modern arithmetization of analysis has had cause to reject. " To Plato are ascribed, among other things, the formulation of the analytic method and the restriction of Euclidean geometry to constructions possible with ruler and compass only. Hankel (Zur Geschichte der Mathematik in Alterthum und Mittelalter, p. 156) ascribes this limitation to Plato, but T. L. Heath (History of Greek Mathematics, I, 288) would place it earlier. » Republic VII. 525-27. " See, in particular, Theaetetus 147-48; Laws 819d-820c. " Parmenides 128 fi. «•Philebas 17 ff. 67 Toeplitz, "Das Verhältnis von Mathematik und Ideenlehre bei Plato." "Ibid., pp. 10-11.

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Plato apparently did not give direct answers to the difficulties involved in incommensurability or in Zeno's paradoxes, although he expressed his opposition to the Pythagorean concepts of infinity, and of the monad as being unity having position," and also to Democritean atomism. He was strongly influenced by both of these schools, but apparently felt that their views were too much the result of sense experience. Plato's criterion of reality was not consistency in experience but reasonableness in thought. For him, as for the Pythagoreans, there was no necessary distinction between mathematics and science; both were the result of deduction from clearly perceived first principles. The Pythagorean monad and the Democritean mathematical atomism, which gave every line a thickness, perhaps appealed too strongly to materialistic sense experience to suit Plato, so that he had recourse to the highly abstract apeiron or unbounded indeterminate. This was the eternally moving infinite of the Ionian philosopher, Anaximander, , ° who had suggested it in opposition to Thales' less subtle assertion that the concrete material element, water, was the basis of all things. According to Plato, the continuum, could better be regarded as generated by the flowing of the apeiron than thought of as consisting of an aggregation (however large) of indivisibles. This view represents a fusion of the continuous and the discrete not unlike the modern intuitionism of Brouwer.61 The infinitely small was apparently not to be reached through a continued subdivision,82 but was to be regarded, perhaps, as analogous to the generative infinitesimal of Leibniz, or the "intensive" infinitely small magnitude which appeared in idealistic philosophy in the nineteenth century. Mathematics has found it necessary to discard both views in making the infinitesimal subordinate to the derivative in the logical foundation of the calculus. However, the notion of the infinitesimal proved very suggestive in the early establishment of the calculus, and, as Newton remarked more than two thousand years later, the application to it of our intuitions of motion removes from the doctrine much of the harshness felt in the mathematical atomism of Democritus and later of Cavalieri. This, however, necessarily led to a loss both of precise logical defi" See T. L. Heath, History of Greek Mathematics, I, 293. 60 Hoppe, "Zur Geschichte der Infinitesimalrechnung," p. 154; see also Milhaud, Les Pküosophes géometres, p. 68. 11 a Helmholtz, Counting and Measuring, pp. xxii-xxiv. Hoppe, op. cit., p. 152.

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29

nition and of clear sensory interpretation, neither of which Plato supplied.4* The belief that mathematics becomes "sterilized by losing contact with the world's work"*4 is widely held but is not easily justified. The conjunction of mathematics and philosophy, as found, for example, in Plato, Descartes, and Leibniz, has been perhaps as valuable in suggesting new advances as has the blending of mathematical and scientific thought illustrated by Archimedes, Galileo, and Newton. The disregard in Platonic thought of any basis in the evidences of sense experience has not unjustly been regarded, from the scientific point of view, as an "unmitigated misfortune." On the other hand, the successful development of his views would have given to mathematics—which is interested solely in relationships which are logically thinkable rather than in those believed to be realized in nature—a flexibility and an independence of the world of sense impressions which were to be essential for the ultimate formulation of the concepts of the calculus. One may therefore say, in a very general sense, that "we know from Plato's own writings that he was thinking out the solution of problems that lead directly to the discovery of the calculus."46 It is, however, altogether too much to assert that "Indeed there are probably only four or five names of mathematical discoverers that stand between Plato on the one hand and Newton and Leibniz, the discoverers of the calculus, on the other hand."44 We shall see that the calculus was the result of a long train of mathematical thought, developed slowly and with great difficulty by very many thinkers. That the doctrines of the continuous and the infinitesimal did not develop along the abstract lines vaguely indicated by Plato was probably the result of the fact that Greek mathematics included no general concept of number,47 and, consequently, no notion of a continuous algebraic variable upon which such theories could logically have been based. Disregard of the abstract idealizations which Plato u Hoppe (op. cit., p. 152) asserts that in Plato one finds the first clear conception of the infinitesimal. It is difficult, however, to perceive on what grounds such a thesis is to be defended. " Hogben, Science for the Citizen, p. 64. Marvin, The History of European Philosophy, p. 142. "Ibid. " Miller, "Mathematical Weakness of Early Civilizations."

30

Conceptions in Antiquity

suggested, but never clearly defined, may also have been due in some measure to the opposition afforded by the inductive scientific views of Aristotle and the Peripatetic School. Aristotelian thought, while not destroying the rigorously deductive character of Greek geometry, may have preserved in Greek mathematics that strong reasonable and matter-of-fact cast which one finds in Euclid and which operated against the early development of the calculus, as well as against the Platonic tendency toward speculative metaphysics. Although Plato did not solve the difficulties which the Pythagoreans and Democritus encountered, he urged their study upon his associates, inveighing against the ignorance concerning such problems which prevailed among the Greeks.68 To Eudoxus he is said to have proposed a number of problems in stereometry which proved to be remarkably suggestive in leading toward the calculus. In this connection the demonstrations which Eudoxus gave of the propositions (previously stated without proof by Democritus) on the volumes of pyramids and cones led to his famous general method of exhaustion and to his definition of proportion. The achievements of Eudoxus are those of a mathematician who was at the same time a scientist, with none of the occult or mystic in him.6' As a consequence, they are based at every point on finite, intuitively clear, and logically precise considerations. In method and spirit the later work of Euclid will be found to owe much more to Eudoxus than to Plato. We have seen that the Pythagorean theory of proportion could not be applied to all lines, many of which are incommensurable, and that the Democritean view of infinitesimals was logically untenable. Eudoxus proposed means by which these difficulties could be avoided. The paths he indicated, in his theory of proportion and in the method of exhaustion, were not the equivalents of our modern conceptions of number and limit, but rather detours which obviated the necessity of using the latter. They were significant, however, in that they made it possible for the Greek mind confidently to pursue its attack upon problems which were to eventuate much later in the calculus. The Pythagorean conception of proportion had been the result of the identification of geometrical magnitudes and integral numbers. « See Laws 819d-820c. •* T. L. Heath, History of Greek Mathematics, I, 323-25; Becker, "Eudoxos-Studien," 1936, p. 410.

Conceptions in Antiquity

31

Two lines, for example, were to each other as the ratio of the (integral) numbers of units in each. With the discovery of the incommensurability of some lines with others, however, this definition could no longer be universally applied. Eudoxus substituted for it another which was more general, in that it did not require two of the terms in the proportion to be (integral) numbers, but allowed all four to be geometrical entities, and required no extension of the Pythagorean idea of number. Euclid70 states Eudoxus' definition as follows: "Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order."71 The theory of proportion thus stated involves only geometrical quantities and integral multiples of them, so that no general definition of number, rational or irrational, is necessary. It is interesting to see that after the development of mathematical analysis, the concept of proportion resembles the arithmetical form of the Pythagoreans rather than the geometrical one of Eudoxus. Even when the ratio is not expressible as the quotient of two integers, we now substitute for it a single number and symbol such as * or e. Although Eudoxus did not, as we do, regard the ratio of two incommensurable quantities as a number,72 nevertheless his definition of proportion expresses the ordinal idea involved in the present conception of real number. The assertion that it is "word for word the same as the general definition of number given by Weierstrass"73 will be found, however, to be incorrect, both literally and in its implications. The formulation of Eudoxus was, on the contrary, a means of avoiding the need of such an arithmetic definition as that of Weierstrass. The method of exhaustion of Eudoxus shows the same abandonment of numerical conceptions which we have seen in his theory of proportion. Length, area, and volume are now carefully defined numerical entities in mathematics. After the time of the Pythagoreans, classic Greek mathematics did not attempt to identify number with 70

Book V, Definition 5. The Thirteen Books of Euclid's Elements, trans, by T. L. Heath, H, 114. Stolz, Vorlesungen iiber allgemeine Arithmetik, I, 94. " Simon, "Historische Bemerkungen ttber das Continuum," p. 387. 71

n

32

Conceptions in Antiquity

geometrical quantities. As a result no rigorous general definitions of length, area, and volume could then be given, the meaning of these quantities being tacitly understood as known from intuition. The question, "What is the area of a circle?" would have had no meaning to the Greek geometers. But the query, "What is the ratio of the areas of two circles?" would have been a legitimate one, and the answer would have been expressed geometrically: "the same as that of squares constructed on the diameters of the circles."74 The fact that squares and circles are incommensurable with each other does not cause any incongruity in the idea of their entering into the same proportion under the generaj definition of Eudoxus; but the proof of the correctness of the proportion requires in this case the comparison of squares with squares and of circles with circles. Obviously the old Pythagorean method of the application of areas cannot be employed in the case of circles, so Eudoxus had recourse to an idea which had been advanced sometime before by Antiphon the Sophist and again a generation later by Bryson. These men had inscribed within a circle a regular polygon, and by successively doubling the number of sides they seem to have hoped to reach a polygon which would coincide with the circle and so "exhaust" its area. It should, however, be borne in mind that we do not know just what Antiphon (and later Bryson) said. The method of Antiphon has been described7' as equivalent at one and the same time to the method of Eudoxus (as given in Euclid XII, 2), and to our conception of the circle as the limit of such an inscribed polygon, but merely expressed in different terminology. This cannot be strictly correct. If Antiphon had considered the process of bisection as carried out to an infinite number of steps, he would not have been thinking in the terms of Eudoxus and Euclid, as we shall see. If, on the other hand, he did not regard the process as continued indefinitely but only as carried out to any desired degree of approximation, he could not have had our idea of a limit. Furthermore, our conception of the limit is numerical, whereas the notions of Antiphon and Eudoxus are purely geometrical. The suggestive idea of Antiphon, however, was adopted by Bryson, who is reputed not only to have inscribed a polygon within the circle 74 n

Cf. Vogt, Der Grenzbegriff in der Eltmeniar^malhematik, p. 42. T. L. Heath, History qf Greek Mathematics, I, 222.

Conceptions in Antiquity

38

but also to have circumscribed one about it as well, saying that the circle would ultimately, as the result of continued bisection, be the mean of the inscribed and circumscribed polygons. Again we do not know exactly what he said, and cannot tell clearly what he meant.7* Interpretations have been advanced77 which would go so far as to see in the work of Bryson the concept of a "Dedekind Cut" or of the continuum of Georg Cantor, but the evidence would hardly appear to warrant such imputations. However, the idea which he suggested was developed by Eudoxus into a rigorous type of argument for dealing with problems involving two dissimilar, heterogeneous, or incommensurable quantities, in which intuition fails to represent clearly the transition from one to the other which is necessary to make a comparison possible. The procedure which Eudoxus proposed has since become known as the method of exhaustion. The principle upon which this method is based is commonly called the lemma, or postulate, of Archimedes, although the great Syracusan mathematician himself ascribed it 78 to Eudoxus and it is not improbable that it had been formulated still earlier by Hippocrates of Chios.7* This axiom (as given in Euclid X, 1) states that, given two unequal magnitudes (neither equal to zero, of course, since for the Greeks this was neither a number nor a magnitude), "if from the greater there be subtracted a magnitude greater than its half, and from that which is left a magnitude greater its half, and if this process be repeated continually, there will be left some magnitude which will be less than the lesser magnitude set out." 80 This definition (in which, of course, any ratio may be substituted in place of one-half) excluded the infinitesimal from all demonstrations in the geometry of the Greeks, although we shall find this banished notion entering occasionally into their thought as an explorative aid. From the fact that, on continuing the process indicated in the axiom of Archimedes, the magnitude remaining can be made as small as we please, the procedure introduced by Eudoxus came much later to be Ibid., I, 224. " See Becker, "Eudoxos-Studien," in particular, 1933, pp. 373-74; ToepliU, "Das Verhältnis von Mathematik und Ideenlehre bei Plato," pp. 31-33. " In his Quadrature of the Parabola. See T. L. Heath, History of Greek Mathematics, I, 327-28. " Hankel, Zur Geschichte der Mathematik in Alterthum und Mittelalter, p. 122. » Euclid, Elements, Heath trans., vol. HI, p. 14.

Conceptions in Antiquity

34

called the method of exhaustion. It is to be remarked, however, that the word exhaustion was not applied in this connection until the seventeenth century,81 when mathematicians somewhat ambiguously and uncritically employed the term indifferently to designate both the ancient Greek procedure and their own newer methods which led immediately to the calculus and which truly "exhausted" the magnitudes. The Greek mathematicians, however, never considered the process as being literally carried out to an infinite number of steps, as we do in passing to the limit—a concept which allows us to interpret the area or volume as truly exhausted, or at least as defined as the limit of the infinite numerical sequence obtained in this manner. There was always, in the Greek mind, a quantity left over (although this could be made as small as desired), so that the process never passed beyond clear intuitional comprehension. A simple illustration will perhaps serve to make the nature of the method clear. The proposition, in Euclid X I I , 2, that the areas of circles are to each other as the squares on their diameters will suffice for this purpose. The substance of this proof is as follows: Let the areas of the circles be A and o, and let their diameters be D and d respectively. If the proportion a : A = d2 : D2 is not true, then let a' : A = d2 : D1, where a' is the area of another circle either greater or smaller than a. If a' is smaller than a, then in the circle of area a we can inscribe a polygon of area p such that p is greater than a' and smaller than a. This follows from the principle of exhaustion (Euclid X, 1)—that if from a magnitude (such as the difference in area between a' and a) we take more than its half, and from the difference more than its half, and so on, the difference can be made less than any assignable magnitude. If P is the area of a similar polygon inscribed in the circle of area A, then we know that p : P = d2 : D2 = a' : A. But since p > a', then P > A, which is absurd, since the polygon is inscribed within the circle. In a similar manner it can be shown that the supposition a' > a likewise leads to a contradiction, and the truth of the proposition is therefore established.82 The method of exhaustion, although equivalent in many respects In particular in Gregory of St. Vincent, Opus geometricum, pp. 739-40. ® Cf. Euclid, Elements, Heath trans., vol. HI, pp. 371-78. 81

Conceptions in Antiquity

35

to the type of argument now employed in proving the existence of a limit in the differential and the integral calculus, does not represent the point of view involved in the passage to the limit. The Greek method of exhaustion, dealing as it did with continuous magnitude, was wholly geometrical, for there was at the time no knowledge of an arithmetical continuum. This being the case, it was of necessity based on notions of the continuity of space—-intuitions which denied any ultimate indivisible portion of space, or any limit to the divisibility in thought of any line segment. The inscribed polygon could be made to approach the circle as nearly as desired, but it could never become the circle, for this would imply an end in the process of subdividing the sides. However, under the method of exhaustion it was not necessary that the two should ever coincide. By an argument based upon the reductio ad absurdum, it could be shown that a ratio greater or less than that of equality was inconsistent with the principle that the difference could be made as small as desired. The argument of Eudoxus appealed at every stage to intuitions of space, and the process of subdivision made no use of such unclear conceptions as that of a polygon with an infinite number of sides—that is, of a polygon which should ultimately coincide with the circle. No new concepts were involved, and the gap between the curvilinear and the rectilinear still remained unspanned by intuition. Eudoxus, however, had most ingeniously contrived to demonstrate—without resort to the logically self-contradictory infinitesimal previously invoked by vague imagination—the truth of certain geometrical propositions requiring a comparison of the curvilinear with the rectilinear and of the irrational with the rational. There is no logical difficulty to be found in the argument used in the method of exhaustion, but the cumbersomeness of its application led later mathematicians to seek a more direct approach to problems in which the application of some such procedure would have been indicated. The method of exhaustion has, most misleadingly, been characterized as "a well-established algorithm of the differential calculus."83 It is indeed true that the problems to which the method was applied were those which led toward the calculus. Nevertheless, it is not incorrect to say that the procedure involved actually directed 83 Simon, "Zur Geschichte und Philosophie der Differentialrechnung," p. 116.

36

Conceptions in Antiquity

attention away from the discovery of an equivalent algorithm, in that it directed attention toward the synthetic form of exposition rather than toward an analytic instrument of discovery." It did represent a conventional type of demonstration, but the Greek mathematicians never developed this into a concise and well-recognized operation with a characteristic notation. In fact the ancients never made the first step in this direction: they did not formulate the principle of the method as a general proposition, reference to which might serve in lieu of the argument by the ubiquitous double reductio ad absurdum." It was largely in connection with the search for some means of simplifying the arguments in the tedious methods of the ancients that the differential and integral calculus was developed in the seventeenth century. To trace this development is the purpose of this essay; but at this point it may not be amiss to anticipate the final formulation to the extent of comparing the nature of the basic concept of the calculus—that of the limit of an infinite sequence—with the view indicated in the method of exhaustion. The limit of the infinite sequence Pi, Pit . . . , Pu, . . . (the terms of which represent, for example, the areas of the inscribed polygons considered in the proposition above) has been defined in the introduction to be the number C, such that, given any positive number e, we can find a positive integer N, such that for n > N it can be shown that | C — PH | < s. The spatial intuition of the method of exhaustion, with its application of areas, unlimited subdivision, and argumentation by a reductio ad absurdum, here gives way to definition in terms of formal logic and number, i. e., of infinite ordered aggregates of the positive integers. The method of exhaustion corresponds to an intuitional concept, described in terms of mental pictures of the world of sensory perception. The notion of a limit, on the other hand, may be regarded as a verbal concept, the explication of which is given in terms of words and symbols—such as number, infinite sequence, less than, greater than—with regard not to any mental visualization, but only to their definition in terms of the primary undefined elements. The limit concept is thus by no means to be considered ineffable; nor does it imply that there is other than Cf. Brunschvicg, Les Étapes de la philosophic mathématique, pp. 157-59. " The Works of Archimedes, ed. by T . L. Heath, p. cxliii. 14

Conceptions in Antiquity

87

empirical experience. It simply makes no appeal to intuition or sensory perception. It resembles the method of exhaustion in that it allows our vague instinctive feeling for continuity to shift for itself in any effort that may be made to picture how the gap between the curvilinear and the rectilinear, or between the rational and the irrational, is bridged, for such an attempt is quite irrelevant to the logical reasoning involved. The limit C is not for this reason to be regarded as a sophistic or inconceivable quantity which somehow nevertheless enters into real relations with other similar quantities, nor is it to be visualized as the last term of the infinite sequence. It is to be considered merely as a number possessing the property stated in the definition. It is to be borne in mind that although adumbrations of the limit idea appear in the history of mathematics in ancient times, nevertheless the rigorous formulation of this concept does not appear in work before the nineteenth century—and certainly not in the Greek method of exhaustion.84 The apparent break, in the mathematical work of Eudoxus, from the metaphysical aspect of Platonism is seen equally clearly8' in the philosophical thought of one who studied under Plato for twenty years and who was known as the "mind of the School."88 Aristotle borrowed freely from the work of his predecessors, with the result that, although not primarily a mathematician, he was familiar with the difficulties and results of Greek mathematics, including the method of exhaustion. He wrote a work (now lost) On the Pythagoreans, discussed at some length the paradoxes of Zeno, mentioned Democritus frequently in mathematics and science (although always to refute him), was intimately familiar with Plato's thought, and was acquainted with the work of Eudoxus. In spite of his competence in mathematics and of his frequent use of geometry in his constructions,8* Aristotle's approach to the problems involved was essentially scientific, in the inductively descriptive sense. Furthermore, for Plato's mathematical intellectualism he substituted a grammatical intuitionism.' 0 " Milhaud (Les Philosophes géomètres, p. 182) would have Eudoxus consider the circles as the limit of a polygon, but this could not have been in the sense in which the term limit is now employed. 87 Cf., in this respect, Jaeger, Aristoteles. • Ross, Aristotle, p. 2. " Enriques, Problems of Science, p. 110. See also Gorland, Aristoteles und die Mathematik, and Heiberg, "Mathematisches zu Aristoteles." * Brunschvicg, Les Étapes de la philosophie mathématique, p. 70.

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Conceptions in Antiquity

Although he realized that the objects of mathematics are not those of sense experience and that the figures used in demonstrations are for illustration only'1 and in no way enter into the reasoning, Aristotle's whole attitude was governed by a strong dependence upon the evidence of the senses, as well as upon logic, and by an aversion to abstraction and extrapolation beyond the powers of sensory perception. As a consequence, he did not think of a geometrical line, as had Plato, as an idea which is prior to, and independent of, experience of the concrete. Neither did he regard it, as does modern mathematics, as an abstraction which is suggested, perhaps, though not in any way defined, by physical objects. He viewed it rather as a characteristic of natural objects which has merely been separated from its irrelevant context in the world of nature. "Geometry investigates physical lines but not qua physical," he said,92 and added: Necessity in mathematics is in a way similar to necessity in things which come to be through the operation of nature. Since a straight line is what it is, it is necessary that the angles of a triangle should equal two right angles.** The decisions which Aristotle rendered on the indivisible, the infinite, and the continuous were consequently those dictated by common sense. In fact, with the exception of Plato's successors in the Academy and, perhaps, of Archimedes, they were those accepted by the body of Greek mathematicians after Eudoxus. Only in the case of the indivisible, however, do Aristotle's views coincide with the present notions in mathematics. Modem science has opposed, modern mathematics upheld, Aristotle in his vigorous denial of the indivisible, physical and mathematical, of the atomic school. Recent physical and chemical theories of the atom have furnished a description of natural phenomena which offers a higher degree of consistency within itself and with sensory impressions than had the Peripatetic doctrine of continuous substantiality. Science has consequently, under Carneades' doctrine of truth, accepted the atom as a physical reality. Modern mathematics, on the other hand, agrees with Aristotle in his opposition to minimal indivisible line segments; not, however, because of any argument from experience, but because it has n

T. L. Heath, History of Greek Mathematics, I, 337. " Physica II. 193b-194a.

• Physica II. 200a.

Conceptions in Antiquity

39

been unable to give a satisfactory definition and logical elaboration of the concept. However, the mathematical indivisible, in spite of the opposition of Aristotle's authority, was destined to play an important part in the development of the calculus, which in the end definitely excluded it. That the concept enjoyed an extensive popularity even in Aristotle's day, Greek logic notwithstanding, is seen by the fact that a Peripatetic treatise formerly ascribed to Aristotle (but now thought to have been written by Theophrastus, or Strato of Lampsacus, or perhaps by someone else), the De lineis insecabilibuswas directed against it. This presents many arguments against the assumption of indivisible lines and concludes that "it conflicts with practically everything in mathematics."' 6 The work may have been composed as an answer to Xenocrates, the successor of Plato in the Academy, who apparently maintained the existence of mathematical indivisibles. I t has been asserted" that neither Aristotle nor Plato's successors understood the infinitesimal concept of their master, and that only Archimedes rose to a correct appreciation of it. It is to be remarked, however, that such an assumption is wholly gratuitous. Plato in his extant works offered no clear definition of the infinitesimal. Archimedes, moreover, made no mention of any indebtedness to Plato in this matter, and, as will be seen, explicitly disclaimed any intention of regarding infinitesimal methods as constituting valid mathematical demonstrations. 87 The opposition of Aristotle to the doctrine of infinitesimals was wholly justified by considerations of logic, although from the point of view of the subsequent development of the calculus the uncritical use of the infinitely small was for a time most fruitful. As in the case of the infinitesimal, so also with respect to the infinite the views of Aristotle constitute excellent illustrations of his abiding confidence in the ultimate interpretability of phenomena in terms of distinctly clear concepts derived from sensory experience.98 The Pythagoreans had regarded space as infinitely divided, and Democritus had likewise spoken of the atoms as infinite in number. Plato, influ« The Works of Aristotle, Vol. VI, Opuscula. * De lineis insecabilibus 970a. ** Hoppe, "Zur Geschichte der Infinitesimalrechnung," p. 152. 87 The Method of Archimedes, ed. by T. L. Heath, p. 17. " Aristotle's view of the infinite has been much discussed. For one of the most recent philosophical discussions of this subject, see Edel, Aristotle's Theory of the Infinite.

40

Conceptions in Antiquity

enced by these views and perhaps also by those of Anaximander and Anaxagoras, had not clearly distinguished the concrete from the abstract." He had held that the infinite was located at the same time in ideas and in the sensible world,100 the line as made up of points being one illustration of this fact. However, no one of Aristotle's predecessors had made quite clear his position with respect to the infinite. Anaxagoras, at least, seems to have realized that it is only the imagination which objects to the infinite and to an infinite subdivision. 101 On the other hand, Aristotle, in adopting the inductive scientific attitude, did not go beyond what is clearly representable in the mind. In consequence he denied altogether the existence of the actual infinite and restricted the use of the term to indicate a potentiality only. 102 His clear distinction between an existent infinity and a potential infinity was the basis of much of the discussion of the Scholastics on the subject and of later controversies on the metaphysics of the calculus. His refusal to recognize the actual infinite was in keeping with his fundamental tenet that the unknowable exists only as a potentiality: that anything beyond the power of comprehension is beyond the realm of reality. Such a methodological definition of existence has led investigators in inductive science to continue to the present time the Aristotelian attitude of negation toward the infinite;103 but such a view, if adopted in mathematics, would exclude the concepts of the derivative and the integral as extrapolations beyond the thinkable, and would, in fact, reduce mathematical thought to the intuitively reasonable. That the Aristotelian doctrine of the infinite was abandoned in the mathematics of the nineteenth century was largely the result of a shift of emphasis from the infinite of geometry to that of arithmetic; for in the latter field assumptions appear to be less frequently dictated by experience. For Aristotle such a change of view would have been impossible, inasmuch as his conception of number was that of the Pythagoreans: a collection of units.104 Zero was not included, of course; nor was "the generator of numbers," the integer one. " T h e " Paul Tannery, Pour l'histoire de la science hellène, pp. 300-5. Brunschvicg, Les Étapes de la philosophie mathématique, p. 67. 101 Paul Tannery, Pour l'histoire de la science hellène, pp. 293-94. ltB Physica LU. 206b. "» Barry, The Scientific Habit of Thought, p. 197. 104 Physica H I . 207b; cf. also Plato, Republic VII. 525e. 100

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smallest number in the strict sense of the word 'number' is two," said Aristotle. 10 * Such a view of number could not be reconciled with the

infinite divisibility

of

continuous magnitude which

Aristotle

upheld so vigorously. When, then, Aristotle distinguished two kinds of (potential) infinite—one in the direction of successive addition, or the infinitely large, and the other in the direction of successive subdivision, or the infinitely s m a l l — w e find the behavior of number to be quite different from that of magnitude: Every assigned magnitude is surpassed in the direction of smallness, while in the other direction there is no infinite magnitude . . . . Number on the other hand is a plurality of "ones" and a certain quantity of them. Hence number must stop at the indivisible. . . . But in the direction of largeness it is always possible to think of a larger number. . . . Hence this infinite is p o t e n t i a l , . . . . and not a permanent actuality but consists in a process of coming to be, like time . . . . With magnitudes the contrary holds. What is continuous is divided ad infinitum, but there is no infinite in the direction of increase. For the size which it can potentially be, it can also actually be. 10 ' In commenting on the view of mathematicians, Aristotle said, In point of fact they do not need the infinite and do not use it. They postulate only that the finite straight line may be produced as far as they wish. . . . Hence, for the purposes of proof, it will make no difference to them to have such an infinite instead, while its existence will be in the sphere of real magnitude.107 H o w well this characterizes Greek geometry can be seen in the method of exhaustion as presented b y E u d o x u s slightly earlier than Aristotle and b y Euclid a little later. T h i s method assumes in the proof only that the bisection can be continued as far as one m a y wish, not carried out to infinity. H o w far it lies from the point of view of modern analysis is indicated b y the fact that the latter has been called " t h e symphony of the infinite." T h a t Aristotle would be unable to cope with the problems of the continuous is perhaps to be expected, both from his view of the infinite and from the lack among the Greeks of an adequate arithmetical point of view. A f t e r having considered place and time in the fourth book of the Physica,

and change in the fifth, Aristotle turned

in the sixth book to the continuous. His account is based upon a m

Physica IV. 220a.

»« Physica i n . 207b.

m

Ibid.

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Conceptions in Antiquity

definition derived from the intuitive notion of the essence of continuous magnitude: "By continuous I mean that which is divisible into divisibles that are infinitely divisible."108 This view was supplemented by a naive appeal to the instinctive feeling of the necessity of a hang-togetherness—of the coincidence of the extremities of the component parts.10' For this reason Aristotle denied that number can produce a continuum,110 inasmuch as there is no contact in numbers.111 Only a generation ago this Aristotelian view had not been entirely abandoned,112 but the mathematical continuum accepted at the present time is defined precisely in terms of the concept of number, or of classes of elements, and of the notion of separation—as in the Dedekind Cut—rather than contact. The nature of continuous magnitude has been found to lie deeper than Aristotle believed, and it has been explained on the basis of concepts which require a broader definition of number than that held during the Greek period. The Aristotelian dictums on the subject were not unfruitful, however, for they led to speculations during the medieval period which in turn aided in the rise of the calculus and the modem doctrine of the continuum. In connection with the study of continuous magnitude, Aristotle attempted also to clarify the nature of motion, criticizing the atomists for their neglect of the whence and the how of movement.113 Although he was quite adept at detecting problems, he failed to make his formulation of these quantitative and was consequently infelicitous in their resolution. In the light of modern scientific method, this lack of mathematical expression gives to his treatment of motion and variability the appearance of a dialectical exercise, rather than of a serious effort to establish a sound basis for the science of dynamics.114 Aristotle's approach to the subject was qualitative and metaphysical. This is evidenced by his definition of motion as "the fulfillment of what exists potentially, in so far as it exists potentially," and by the further remark, "We can define motion as the fulfillment of the movable qua movable."116 We shall find this qualitative explanation of motion—the result of the striving of a body to become actually what it is potentially—involved in less teleological forms in the idea 106 l0 Pkysica VI. 232b. » Physica VI. 231a; cf. also Categorías 5a. m "" Metaphysial 1075b and 1020a; Categoriae 4b. Metaphysial 1085a. m See Merz, A History of European Thought in the Nineteenth Century, n , 644. m Metaphysica 985b and 1071b. m U4 Cf. Mach, The Science of Mechanics, p. 511. Physica m , 201a-202a.

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of impetus developed by the Scholastics, in Hobbes' explanation of velocity and acceleration in terms of a conatus, and even in metaphysical interpretations of the infinitesimal of the calculus as an intensive quantity—that is, as a "becoming" rather than a "being." In this respect Aristotle's work may have encouraged the elaboration of notions leading toward the derivative. However, his influence was in another sense quite adverse to the development of this concept in that it centered attention upon the qualitative description of the change itself, rather than upon a quantitative interpretation of the vague instinctive feeling of a continuous state of change invoked by Zeno. The calculus has shown that the concept of continuous change is no more free from that of the discrete than is the numerical continuum, and that it is logically to be based upon the latter, as is also the idea of geometrical magnitude. As long as Aristotle and the Greeks considered motion continuous and number discontinuous, a rigorous mathematical analysis and a satisfactory science of dynamics were difficult of achievement. The treatment of the infinite and of continuous magnitude found in the Physica of Aristotle has been regarded as presenting the appearance of a veritable introduction to a treatise on the differential calculus.116 Such a view, however, is seen to be most unwarranted, inasmuch as Aristotle expressed his unqualified opposition to the fundamental idea of the calculus—that of an instantaneous rate of change. He asserted that "Nothing can be in motion in a present. . . . Nor can anything be at rest in a present."117 This point of view necessarily operated against the mathematical representation of the phenomena of change and against the development of the calculus. Aristotle's denial of instantaneous velocity, as realized in the world described by science, is, to be sure, in conformity with the recognized limitations of sensory perception. Only average velocities, — , are At recognizable in this sense. In the world of thought, on the other hand, it has been found possible—through the calculus and the limit concept—to give a rigorous quantitative definition of instantaneous velocity, —. Aristotle, however, in conformity with a view widely acdt ut

Moritz Cantor, "Origines du calcul infinitésimal," p. 6.

m

Physica VI. 234a.

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cepted at the time, regarded mathematics as a pattern of the world known through the senses and consequently did not foresee such a possibility. The failure of Aristotle to distinguish sharply between the worlds of experience and of mathematical thought resulted in his lack of clear recognition of a similar confusion in the paradoxes of Zeno. Aristotle refuted the arguments in the stade and the arrow by an appeal to sensory perception and the denial of an instantaneous velocity. Modern mathematics, on the other hand, has answered them in terms of thought alone, based on the concept of the derivative. In the same manner Aristotle resolved the paradoxes in the dichotomy and the Achilles by the curt assertion, suggested by experience, that although one cannot traverse an infinite space in finite time, it is possible to cover an infinitely divided space in finite time because of the infinite divisibility of the latter." 8 Mathematics has, of course, given the solution of the difficulties in terms of the abstract concept of converging infinite series. In a certain metaphysical sense this notion of convergence does not answer Zeno's argument, in that it does not tell how one is to picture an infinite number of magnitudes as together making up only a finite magnitude; that is, it does not give an intuitively clear and satisfying picture, in terms of sense experience, of the relation subsisting between the infinite series and the limit of this series. If one demands that Zeno's paradoxes be answered in terms of our vague instinctive feeling for continuity—as essentially different from the discrete—no answers more satisfying than those of Aristotle (to whom we owe also the statement of the paradoxes, since we do not have Zeno's words) have been given. The unambiguous demonstration that the difficulties implied by the paradoxes are simply those of visualization and not those of logic was to require more precise and adequate definitions than any which Aristotle could furnish for such subtle notions as those of continuity, the infinite, and instantaneous velocity. Such definitions were to be given in the nineteenth century in terms of the concepts of the calculus; and modern analysis has, upon the basis of these, clearly dissented from the Aristotelian pronouncements in this field. The views of Aristotle are not on this account to be regarded— "» Physica VI. 239b-240a.

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45 11

as is all too frequently and uncritically maintained *—as gross misconceptions which for two thousand years retarded the advancement of science and mathematics. They were, rather, matured judgments on the subject which furnished a satisfactory working basis for later investigations which were to result in the science of dynamics and in the mathematical continuum. Nevertheless, there is apparent in the work of Aristotle the cardinal weakness of Greek logic and geometry : a naïve realism which regarded thought as a true copy of the external world.120 This caused him to place too ingenuous a confidence in certain instinctive feelings with respect to continuous magnitude and to seek, of all possible representations, that which presented the greatest plausibility in the light of sensory experience, rather than that which offered the widest consistency in thought. It has been said121 that the fifth book of Euclid's Elements and the logic of Aristotle are the two most unobjectionable and unassailable treatises ever written. The two men were roughly contemporaries: Aristotle lived from 384 to 322 B. C. ; Euclid's birth has been placed at about 365 B. C. and the composition of the Elements may be accepted as between 330 and 320 B. C.122 There is also a marked similarity between the Aristotelian apodictic and the mathematical method built up by Euclid.123 Although Euclid was probably taught by the pupils of Plato,124 the influence of the sober, hard-headed, scientific thought illustrated by Eudoxus and Aristotle must have predominated over the more abstract, speculative, and even mystical trend seen in the immediate successors of Plato to the leadership of the Academy and carried to excess by later Neoplatonists. There is in Euclid none of the metamathematics which played such a prominent part in Plato's thought, nor do metaphysical speculations on mathematical atomism enter. Mathematics was regarded by Euclid neither as a necessary form of cosmological intelligibility, nor as a mere tool of See, for example, Mayer, "Why the Social Sciences Lag behind the Physical and Biological Sciences"; cf. also Toeplitz, "Das Verhältnis von Mathematik und Ideenlehre bei Plato." m Enriques, The Historic Development of Logic, p. 25. 121 By Augustus De Morgan. See Hill, "Presidential Address on the Theory of Proportion." m See Vogt, "Die Lebenzeit Euklids." 123 Brunschvicg, Les Étapes de la philosophie mathématique, pp. 84-85. m T. L. Heath, History of Greek Mathematics, I, 356.

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Conceptions in Antiquity

pragmatic utilitarianism. For him it had entered the domain of logic, and in this connection Proclus tells us that Euclid subjected to rigorous proofs what had been negligently demonstrated by his predecessors.1" Nevertheless, the Elements retained the realism which was so clearly apparent in Aristotelian logic.126 Although Aristotle had rejected Plato's doctrine of ideas, he had retained a belief in a natural order of science and in the necessary character of principles. This latter confidence was adopted likewise by Euclid. Greek geometry was not formal logic, made up of hypothetical propositions, as mathematics largely is today; but it was an idealized picture of the world of actuality. Just as Aristotle seems not to have clearly recognized the tentative character of scientific knowledge (thus leaving himself open to the attacks of the Skeptics), so also he failed to appreciate that although the conclusions drawn by mathematics are necessary inferences from the premises, nevertheless the latter are quite arbitrarily selected, subject only to an inner compatibility. Aristotle considered hypotheses and postulates as statements which are assumed without proof, but which are nevertheless capable of demonstration.127 Although he admitted that "we must get to know the primary premises by induction" (rather than by pure intellection, as Plato had believed), he maintained that "since except intuition nothing can be truer than scientific knowledge, it will be intuition that apprehends the primary premises," and primary premises are therefore "more knowable than demonstrations."128 The Euclidean view was similar to the Peripatetic attitude in giving to geometry the characteristic form of logical conclusions from necessary postulates. As such, it excluded any notions the nature of which was not clearly and compellingly "felt" through intuition. The infinite was never invoked in the demonstrations, true to Aristotle's statement that it was unnecessary, its place being taken by the method of exhaustion which had been developed by Eudoxus. The limitation of the concept of number to that of positive integers apparently was 115 Cf. Proclus Diadochus, In primum Eudides elementorum librurn commentariorum . . . ibri IIII, p. 43. m Enriques, The Historic Development of Logic, p. 25; cf. also Burtt, Metaphysical Foundations of Modern Physical Science, p. 31. 117 Analytica posteriora I. 76b. m Analytiat posteriora II. 100b; cf. also Brunschvicg, Les Étapes de la philosophic mathématique, pp. 86-93.

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47

continued, a broader view being made unnecessary by the Eudoxian theory of proportion.1" For Euclid ratio was not a number in the abstract arithmetical sense (and in fact it did not become so until the time of Newton),00 and the treatment of the irrational in the Elements is completely geometrical. Furthermore, the axioms, postulates, and definitions of Euclid are those suggested by common sense, and his geometry never loses contact with spatial intuition.131 His premises are the dictates of sensory experience, much as Aristotle's science may be characterized as a glorification of common sense. Such purely formal, logical concepts as those of the infinitesimal and of instantaneous velocity, of infinite aggregates and the mathematical continuum, are not elaborated in either Euclidean geometry or in Aristotelian physics, for common sense has no immediate need for them. The ideas which were to lead to the calculus had not in Euclid's time reached a stage at which a logical basis could have been afforded; mathematics had not attained the degree of abstraction demanded for symbolic logic. Although the origin of the notions of the derivative and the integral are undoubtedly to be found in our confused thought about variability and multiplicity, the rigorous formulation of the concepts involved, as we shall find, demanded an arithmetical abstraction which Euclid was far from possessing. Even Newton and Leibniz, the inventors of the algorithmic calculus, did not fully recognize the need for it. The logical foundations of the calculus are much further removed from the vague suggestions of experience—much more subtle—than those of Euclidean geometry. Since, therefore, the ideas of variability, continuity, and infinity could not be rigorously established, Euclid omitted them from his geometry. The Elements are based on "refined intuition,"132 and do not allow free scope to the "naïve intuition" which was to be especially active in the genesis of the calculus in the seventeenth century.133 From the point of view of the development of the calculus, there™ See Stolz, Vorlesungen über allgemeine Arithmetik, I, 94; cf. also Schubert, "Principes fondamentaux de l'arithmétique," pp. 8-9. uo See Newton, Opera omnia, I, 2. U1 Cf. Barry, The Scientific Habit of Thought, pp. 215-17. m As Felix Klein (The Evanston Colloquium Lectures on Mathematics, pp. 41-42) so aptly puts it. m Ibid.

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Conceptions in Antiquity

fore, the Elements of Euclid show an uninteresting inflexibility of rigor, discouraging to the growth of such new speculations and discoveries. The work of Euclid represents the final synthetic form of all mathematical thought—the elaboration by deductive reasoning of the logical implications of a set of premises. Back of his geometry, however, stood several centuries of analytical investigation, carried out often on the basis of empirical research, or on uncritical intuition, or, not infrequently, on transcendental speculation. It was to be largely from indagation of a similar type, rather than from the rigorously precise thought of Euclid, that the development of the concepts of the calculus was to proceed. This in its turn was necessarily to give way, in the nineteenth century, to a formulation as eminently deductive—albeit arithmetic rather than geometric—as that found in the Elements. The greatest mathematician of antiquity, Archimedes of Syracuse, displayed two natures, for he tempered the strong transcendental imagination of Plato with the meticulously correct procedure of Euclid. He "gave birth to the calculus of the infinite conceived and brought to perfection successively by Kepler, Cavalieri, Fermât, Leibniz, and Newton,"1*4 and so made the concepts of the derivative and the integral possible. In the demonstration of his results, however, he adhered to the clearly visualized details of the Eudoxian procedure, modifying the method of exhaustion by considering not only the inscribed figure but the circumscribed figure as well. The deductive method of exhaustion was not a tool well adapted to the discovery of new results, but Archimedes combined it with infinitesimal considerations toward which Democritus and the Platonic school had groped. The freedom with which he handled these is shown most clearly in the treatise to which we have already referred, the Method.™

This work, addressed to Eratosthenes the geographer, astronomer, and mathematician of Alexandria, was lost and remained largely unm Chasles, Aperçu historique sur l'origine et le développement des méthodes en gtomitrie, p. 22. m For the works of Archimedes in general, see Heiberg, Arckimedis opera omnia and T. L. Heath, The Works of Archimedes. For Archimedes' Method, see T. L. Heath, The Method of Archimedes, Recently Discovered by Heiberg; Heiberg and Zeuthen, "Eine neue Schrift des Archimedes"; and Smith, "A Newly Discovered Treatise of Archimedes."

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known until rediscovered in 1906. In it Archimedes disclosed the method which is presumably that which he employed in reaching many of his conclusions in problems involving areas and volumes. Realizing that it is advantageous to have a preliminary notion of the result before carrying through a deductive geometrical demonstration, Archimedes employed for this purpose, in conjunction with his law of the lever, the idea of a surface as made up of lines. For example, he showed that the truth of the proposition that a parabolic segment is i the triangle having the same base and vertex (the vertex of the segment being taken as the point from which the perpendicular to the base is greatest) is indicated by the following considerations from

FIGURE 2 mechanics.136 In the diagram given in Figure 2, in which V is the vertex of the parabola, BC is tangent at B, BD = DP, and X is any point on AB, we know from the properties of the parabola that for , , • XX'" AB BD DP „ ,„ any position of X we have the ratio = —— = = . But XX' AX DX' DX" X" is the center of gravity of XX'", so that from the law of the lever we see that XX', if brought to P as its midpoint, will balance XX'" in its present position. This will be true for all positions of X on AB. Inasmuch as the triangle ABC consists of the straight lines XX'" in this triangle, and since the parabolic segment A VB is likewise made u

' See T. L. Heath, The Method of Archimedes, Proposition I, pp. 15-18.

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Conceptions in Antiquity

up of the lines XX', we can conclude that the triangle ABC in its present position will be in equilibrium at D with the parabolic segment when this is transferred to P as its center of gravity. But the center of gravity of ABC is on BD and is 3 the distance from D to B, so that the segment A VB is 3 the triangle ABC, or 7 the triangle A VB. This method of Archimedes indicates an anticipation of the use of the concept of the indivisible which was to be made in the fourteenth century and which, when developed again more freely in the seventeenth century, was to lead directly to the procedures of the calcuius. The basis of the method is to be found in the assumption of Archimedes that surfaces may be regarded as consisting of lines. We do not know in precisely what sense he intended this to be understood, for he did not speak of the number of elements in each figure as infinite, but said rather that the figure is made up of all the elements in it. That he probably thought of them as mathematical atoms is indicated not only by this manner of expression, but also by the highly suggestive fact that he was led to many new results by a process of balancing, in thought, elements of dissimilar figures, using the principle of the lever precisely as one would in weighing mechanically a collection of thin laminae or material strips. Using this heuristic method, Archimedes was able to anticipate the integral calculus in achieving a number of remarkable results. He discovered, among other things, the volumes of segments of conoids and cylindrical wedges and the centers of gravity of the semicircle, of parabolic segments, and of segments of a sphere and a paraboloid.137 However, to assert that here "for the first time one can correctly speak of an integration"138 is to misinterpret the mathematical process known by this name. The definite integral is defined in mathematics as the limit of an infinite sequence and not as the sum of an infinite number of points, lines, or surfaces.139 Infinitesimal considerations, similar to those in the Method, were at a later period to furnish perhaps the strongest incentive to the development of the calculus, but, w See The Works of Archimedes, Chap. VII, "Anticipations by Archimedes of the Integral Calculus"; also Method. u * Hoppe, "Zur Geschichte der Inlinitesimalrechnung," p. 154; cf. also p. 155. m T. L. Heath (The Method of Archimedes, pp. 8-9) correctly points out that the method here used is not integration; but he gratuitously imputes to Archimedes the concept of the differential of area.

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as Archimedes realized, they lacked in his time all basis in rigorous thought. This they continued to do until the concepts of variability and limit had been carefully analyzed. For this reason Archimedes considered that this method merely indicated, but did not prove, that the result is correct.140 Archimedes employed his heuristic method, therefore, simply as an investigation preliminary to the rigorous demonstration by the method of exhaustion. It was not a generous gesture that led Archimedes to supplement his "mechanical method" by a proof of the results in the rigorous manner of the method of exhaustion; it was, rather, a mathematical necessity. It has been asserted that Archimedes' method "would be quite rigorous enough for us today, although it did not satisfy Archimedes himself."141 Such an assertion is strictly correct only if we ascribe to him our modern doctrines on number, limit, and continuity. This ascription is hardly warranted, inasmuch as Greek geometry was concerned with form, rather than with variation. It was, as a result, necessarily unable to frame a satisfactory definition of the infinitesimal, which of necessity was to be regarded as a fixed quantity rather than as an auxiliary variable. Archimedes was probably well aware of the lack of any sound basis for his method and for this reason recast all of his analysis by infinitesimals in the orthodox synthetic form, much as Newton was to do almost nineteen hundred years later after the methods of the calculus had been discovered but still lacked adequate foundation. The suggestive analysis of the problem of determining the area of a parabolic segment had been given by Archimedes in the Method. However, formal proofs (both mechanical and geometrical) of the proposition were carried out by the method of exhaustion in another treatise, the Quadrature of the Parabola.142 In these proofs Archimedes followed his illustrious predecessors in omitting all reference to the infinite and the infinitesimal. In the geometrical demonstration, for example, he inscribed within the parabolic segment a triangle of area A, having the same base and vertex as the segment. Then within each T. L. Heath, History of Greek Mathematics, II, 29. T. L. Heath, The Method of Archimedes, p. 10. l a See The Works of Archimedes, and T. L. Heath, History of Greek Mathematics, II, 8591. A good adaptation of the geometrical proof is given in Smith, History of Mathematics, H, 680-83. 140 141

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Conceptions in Antiquity

of the two smaller segments having the sides of the triangle as bases, he similarly inscribed triangles. Continuing this process, he obtained a series of polygons with an ever-greater number of sides, as illustrated (fig. 3). He then demonstrated that the area of the nth such polygon + . . . + > where A was given by the series A ( l + | + is the area of the inscribed triangle having the same vertex and base as the segment. The sum to infinity of this series is $A, and it was probably from this fact that Archimedes inferred that the area of the parabolic segment was also $A. liS However, he did not state the argument in this maimer. Instead of finding the limit of the infinite series, he found the sum of » terms and added the remainder, using the equality A ( l + i - f ^

+

. - . + ^

+

i . ^ )

As the number of terms becomes greater, the series thus "exhausts" iA only in the Greek sense that the remainder, ^ can be made as small as desired. This is, of course, exactly the method of proof for the existence of a limit,144 but Archimedes did not so interpret the argument. He did not express the idea that there is no remainder in the limit, or that the infinite series is rigorously equal to $ A . i a Instead, he proved, by the double reductio ad absurdum of the method of exhaustion, that the area of the parabolic segment could be neither greater nor less than $A. In order to be able to define i A as the sum of the infinite series, it would have been necessary to develop the general concept of real number. Greek, mathematicians did not possess this, so that for them there was always a gap between the real (finite) and the ideal (infinite). "»T. L. Heath, "Greek Geometry with Special Reference to Infinitesimals." 144 As Miller pointed out in "Some Fundamental Discoveries in Mathematics." 144 The Works qj Archimedes, p. cxliii.

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It is not strictly correct, therefore, to speak of Archimedes' geometrical procedure as a passage to the limit, for the essential part of the definition of a limit is the infinite sequence.14* Inasmuch as he did not invoke the limit concept, it is hardly correct to say that in finding the sum of such series Archimedes answered in a very explicit and definite manner some of the difficult questions raised by Zeno, and that "These difficulties were completely solved by the Greek mathematicians, and further serious arguments along this line seem to be based upon ignorance or perversity."147 The notion of the limit of an infinite series is essential for the clarification of the paradoxes; but Greek mathematicians (including Archimedes) excluded the infinite from their reasoning. The reasons for this ban are obvious: intuition could at the time afford no clear picture of it, and it had as yet no logical basis. The latter difficulty having been removed in the nineteenth century and the former being now considered irrelevant, the concept of infinity has been admitted freely into mathematics. The related limit concept is now invoked in the explication of the paradoxes, as well as in a simplification of Archimedes' long indirect demonstrations. The series given above is not the only one found in Archimedes' work. In determining, by the method of exhaustion, the volume of a segment of a paraboloid of revolution, he was led to investigations of a similar nature. A consideration, in some detail, of the application of the method which Archimedes here made148 may be desirable at this point, in order to bring out clearly the general character of his procedure and to point out to what a remarkable extent it resembles that used in the integral calculus, even though Archimedes did not explicitly employ the limit concept. Archimedes first circumscribed about the solid ABC (which he called a conoid) the cylinder ABEF (fig. 4) having the same axis, CD, as has the paraboloidal segment. He then divided the axis into n equal parts of length h and through the points of division passed planes parallel to the base. On the sections of the paraboloid thus formed he constructed inscribed and circumscribed cylinder frusta, as 141 C. R. Wallner, "Ober die Entstehung des Grenzbegriffes," p. 250. See also Hankel, "Grenze"; and Wieleitner, Die Geburt der Modernen Mathematik, II, 12. 10 Miller, "Some Fundamental Discoveries," p. 498. 148 On Conoids and Spheroids, Propositions 21, 22; The Works of Archimedes, pp. 131-33.

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shown in the figure. He was then able to establish the equivalent of the proportions: Cylinder ABEF tfh Inscribed figure h + 2k + . . . + (* - \)h , and Cylinder ABEF = tMt Circumscribed figure h + 2h + . . . + nk Archimedes had previously shown149 (using a method cast in geometrical form but otherwise much like that ordinarily employed in elementary algebra to determine the sum of an arithmetic progression) that k + 2h + . . . + («l)h < \n^k and that k + 2k + . . . +

concept and, allowing the series h + 2h + 3h+ would conclude that

lim

» —•

I\h +

^ 2h +

3k +

.. .to become infinite, — — 7 ) = 2. Not so . . . +

nh)

Archimedes. Instead of doing this, he showed that the proportions above may be written, as a result of these inequalities, Cylinder ABEF — > 2- and. Cylinder ABEF , . < 2r-

inscribed figure 1 circumscnbed figure 1 Now by the principle of exhaustion and the usual reductio ad absurdum, he concluded that the paraboloidal segment can be neither greater than, nor less than, half the cylinder ABEF. In the argument of Archimedes, the logical validity of a conclusion lm

In Proposition 10 of the work On Spirals. See Works 0/ Archimedes, pp. 163-65.

Conceptions in Antiquity

55

based on the numerical concept of the limit of an infinite series was not admitted, but was replaced by one based on the rigorous geometrical method of exhaustion. I t has been said that the differences in the methods of infinitesimals, of exhaustion, and of limits are felt to be more in the words than in the ideas;150 but such an assertion may lead to serious misinterpretation. The methods are, of course, interrelated, and in consequence they lead to identical results; but the points of view are distinctly different, as we have seen. Although in a broad sense the procedures of Archimedes may be considered as "practically integrations," 151 or even as representing in a general way "an integration process,"152 it is indeed far from correct to speak of any one of them as a "veritable integration" 1 " or as "the equivalent of genuine integration." 1 " The definite integral requires for its correct formulation an appreciation of the notions of variability and funcn

tionality, the formation of the characteristic sum 2 /(x,)Ax { , and the t=i

application of the concept of the limit of the infinite sequence obtained from this sum by allowing n to increase indefinitely as Ax,- becomes indefinitely small. These essential aspects of the integral are, of course, at no place indicated in the work of Archimedes, for they were extrinsic to the whole of Greek mathematical thought. The series k + 2h + 3h + . . . , employed by Archimedes in the proposition above, was to figure prominently in the gropings toward the calculus in the seventeenth century, so that it may be well to point out that this geometrical demonstration is in broad outline equivalent to performing the integration indicated by J^xdx. In determining that the area bounded by the polar axis and by one ad

turn of the spiral P = — is | that of the circle of radius a,155 Archimedes 2r

had occasion to make a similar calculation, equivalent to evaluating f£x2dx. This was not done directly and arithmetically by determining Hm 11111 " -

A2 -

\

+

vW

—

+

. . . +

n(nhY

v(nh)*\

,

,

' ] to be 1, as he might easily I

See Milhaud, Nouvelles iludes, p. 149. T. L. Heath, History of Greek Mathematics, I I , 3. u 2 Karpinski, " I s There Progress in Mathematical Discovery?" p. 48. "* Zeuthen, Geschichle der Mathematik im Alterturn und Mittdalter, p. 181. 144 The Works of Archimedes, p. cliii; cf. also p. cxliii. » On Spirals, Proposition 24, The Works of Archimedes, pp. 178-82. 140 151

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Conceptions in Antiquity

have done had the Greeks not interdicted the infinite, but indirectly and geometrically by coupling the inequalities . A» + (2h)> + . . . + [ ( » _ 1)*]» k* + (2hY + . . . + (nh)* ^ . —— > 4 and — < n(nh¥

n{nhY

with the proof by the method of exhaustion, in a manner similar to that employed in the proposition on the paraboloidal segment. Archimedes may have known the equivalent result for the sum of cubes as well, and much later the Arabs extended his work to include also the fourth powers. In the seventeenth century at least a half dozen mathematicians —Cavalieri, Torricelli, Roberval, Fermat, Pascal, and Wallis—were to extend (all more or less independently) this work of Archimedes still further by the determination of fyfdx for yet other values of n, and in this way to point immediately to the algorithm of the calculus. The methods which these men were to use were not, in general, the careful geometric procedures found in the propositions of Archimedes, but were to be based on ideas of indivisibles and of infinite series—notions which the decline of the idea of mathematical rigor during the intervening years was to make more acceptable to mathematicians, even though they were to be at that time no less impeachable than in Archimedes' day. We have seen that Greek geometry was concerned largely with form rather than with variation, so that the function concept was not developed. That motion was nevertheless occasionally invoked in mathematics is indicated by Plato's suggestion that a line is generated by a moving point, as well as by the fact that certain special curves, described by a double motion, had been discussed even before Plato's time. The most famous of such curves was perhaps the quadratrix of Hippias, the Sophist. Archimedes may have been influenced by Hippias' idea when he defined his spiral as the locus of a point which moved with uniform radial velocity along a line, while the line in turn revolved uniformly about one of its end points which is kept fixed.166 In the further study of this curve Archimedes was led, in attempting to determine its tangent, to one of the few considerations corresponding to the differential calculus to be found in Greek geometry. In conformity with the static geometry of design, a tangent to a circle The Works of Archimedes, p. 165.

Conceptions in Antiquity

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17

had been defined by Euclid ' as a line touching the circle at only one point, and this definition was extended by Greek geometers to apply to other curves as well. There is also ascribed to the ancients1** the definition, following the suggestion contained in a proposition of Euclid,168 of a tangent as a line touching a curve and such that in the space between the straight line and the curve no other straight line can be interposed. These definitions were, of course, of restricted applicability and did not, in general, suggest a method of procedure for drawing tangents. Although Archimedes did not offer a more satisfactory definition, he appears, nevertheless, to have employed, for the determination of the tangent to his spiral, a method suggestive of a more general point of view. As in quadratures he had used considerations from the science of statics, so here in the problem of tangents Archimedes appears to have had recourse to a representation derived from kinematics. It seems likely, although Archimedes did not thus express the idea, that he found the tangent to the spiral p = ad through a determination of the instantaneous direction of motion of the point P, by which it is traced.160 This he probably did by applying to the motions by which the spiral may be generated the parallelogram of velocities, the principle of which had been perceived by the Peripatetics.161 The motion of P may be regarded as compounded of two resultant motions: one with a radial velocity of constant magnitude VT directed along the line OP (see fig. 5), and the other in a direction perpendicular to this and having a magnitude which is given by the variable product, Va, of the distance OP and the uniform speed of rotation. Inasmuch as the distance OP and the speeds are given, the parallelogram (in this case a rectangle) of velocities can be constructed and the direction of the resultant velocity, and therefore also the tangent PT, determined. The determination by Archimedes of the tangent to the spiral has been characterized as "a differentiation,"162 or as "corresponding to Book HI, Definition 2, in T. L. Heath ed., II, 1. See, for example, Comte, The philosophy of Mathematics, pp. 108-10. Book i n , Proposition 16, in T. L. Heath ed., II, 37. 1,0 T. L. Heath, History of Greek Mathematics, II, 556-61. 181 See Mechanica X X X I H . 854b-855a in The Works of Aristotle, Ross ed., Vol. VI, Opuscula; cf. also Duhem, Les Origines de la statique, H, 245; and Mach, The Science of Mechanics, p. 511. Simon, "Zur Geschichte und Philosophie der Differentialrechnung," p. 116. 168

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our use of the differential calculus."1" Such a designation, however, is hardly justified. Apparently, Archimedes made no effort to develop the idea here involved into a uniform method of attack upon the problem of tangents to other curves. When, in the seventeenth century, the method again appeared in the work of Torricelli, Roberval, Descartes, and Barrow, the scope of its applicability was somewhat extended; but only with the method of fluxions of Newton was there presented an algorithmic procedure for determining from the equation of any curve a pair of generating motions from which the tangent might be found. There was in Greek geometry no idea of a curve as

O

T

FIGURE 5 corresponding to a function, nor was there a satisfactory definition of a tangent in terms of the limit concept. There was therefore in the thought of Archimedes no anticipation of the realization that the geometrical notion of tangency is to be based upon the function concept and upon the numerical idea of a limit, i. e., upon the expression which furnishes the basis for the differential calculus. There is in the whole of Greek mathematics no clear recognition of the need for the limit concept, both for the determina1U

T. L. Heath, Hislot y of Greek Mathematics, II, 557.

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59

tion of curvilinear areas and of tangents to curves, and even for the very definition of these ideas which intuition vaguely suggests. It is therefore incorrect to impute to Archimedes the ideas expressed in the integral and the derivative. These notions were not part of Greek geometry. Nevertheless, the problems and methods of Archimedes furnished probably the strongest incentive to the later development of such ideas, for they were leading in precisely this direction. The work of Archimedes so strongly suggests the newer methods of analysis that in the seventeenth century Torricelli and Wallis hazarded the opinion that the ancient Greek mathematicians had deliberately concealed under their synthetic demonstrations the analytic devices by which they had been led to their discoveries.1*4 Through the discovery of the Method of Archimedes, the assumption of the existence of such methods has been proved correct, but the failure of ancient geometers further to elaborate these in their works is not to be regarded as indicating an intent to deceive. Although the notions of instantaneous velocity and of the infinitely small were accepted—all too uncritically—by Torricelli, Wallis, and their contemporaries, these ideas were not considered by Greek thinkers as admissible in mathematics. In the seventeenth century, however, the infinitesimal and kinematic methods of Archimedes were made the basis of the differential and the fluxionary forms of the calculus. Although Leibniz and Newton were thus to admit into the calculus the notions of the infinitely small and of instantaneous velocity, these were to remain open to criticism even after that—in fact, until, in the nineteenth century, the basic concept, that of the derivative, had been carefully defined. After the time of Archimedes, the trend of Greek geometry was toward applications, rather than toward new theoretical developments.166 No Greek mathematician approached nearer to the calculus than had Archimedes. His successors—Hipparchus, Heron, Ptolemy, and others—turned to various mathematical sciences, such as astronomy, mechanics, and optics. Nevertheless, the infinitesimal considerations of Archimedes were not forgotten by the Greek geometers of later times. Toward the end of the third century of our era, in a 146

T. L. Heath, History of Greek Mathematics, n , 21,557; Torricelli, Opere, I (Part 1), 140T. L. Heath, History of Greek Mathematics, II, 198.

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period which is regarded as one of decline in mathematics, the geometer Pappus not only displayed a familiarity with these methods, but was able also to add a new result—known as the Pappus theorem— to the work of Archimedes on centers of gravity.1*6 However, further significant advances in geometrical method were to be dependent upon certain broad changes in other branches of mathematics. One of these was the development of a more highly elaborated abstract symbolic algebra; another was the introduction into algebra and geometry of the notion of variation—of variables and functionality. The Arithmetic of Diophantus, which represents the highest development of Greek algebraical thought, was really theoretical logistic,1'7 rather than generalized arithmetic or the study of certain functions of variables. In it the Peripatetic basis in logic is stronger than the Platonic ontological conception of mathematics. Only one unknown was introduced in this work, and the only solutions which were accepted as having meaning were those which, in accordance with Aristotelian tradition, could be expressed as the quotient of integers. Irrational and imaginary numbers were not recognized.1" The Arithmetic of Diophantus represents a "synocopated algebra," in that abbreviations for certain recurring quantities and operations are systematically introduced in it. However, in order that his work might be associated with geometrical results and later serve as the suitable basis for the calculus, it had to be made more completely symbolic, the concept of number had to be generalized, and the ideas of variable and function had to be introduced. During the Middle Ages the interest of the Hindus and the Arabs in algebraic development, and the attack by the Scholastic philosophers upon the problems offered by the continuum, were to supply in some measure the background required for these changes. When, therefore, in the sixteenth and seventeenth centuries, the classical work of Archimedes was developed into the methods constituting the calculus, the advance was made along lines suggested by the traditions built up during the medieval period. ut We cannot be sure, however, that the theorem was an original contribution on the part of Pappus or that he possessed a proof of it. See Pappus of Alexandria, La Collection mathématique, trans, by Ver Eecke; cf. also Weaver, "Pappus." 1,7 Jakob Klein, "Die griechische Logistik und die Entstehung der Algebra"; see also Tannery, Pour la science hellène, p. 405. T. L. Heath, History of Greek Mathematics, I, 462.

III. Medieval Contributions

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HE MOOT points as to the origin and antiquity of Hindu mathematics are not immediately pertinent in a precis of the development of the derivative and the integral; for these concepts depend on certain logical subtleties, the significance of which appears to have surpassed the appreciation, or at least to have escaped the interest, of the early Indian mathematicians. The Hindus apparently were attracted by the arithmetical and computational aspects of mathematics,1 rather than by the geometrical and rational features of the subject which had appealed so strongly to the Hellenic mind. Their name for mathematics, ganita, meaning literally the "science of calculation,"2 well characterizes this preference. They delighted more in the tricks that could be played with numbers than in the thoughts the mind could produce, so that neither Euclidean geometry nor Aristotelian logic made a strong impression upon them. The Pythagorean problem of the incommensurable, which was of intense interest to Greek geometers, was of little import to Hindu mathematicians, who treated rational and irrational quantities, curvilinear and rectilinear magnitudes indiscriminately.* With respect to the development of algebra, this attitude occasioned perhaps an incidental advance, since by the Hindus the irrational roots of quadratics were no longer disregarded, as they had been by the Greeks, and since to the Hindus we owe also the immensely convenient concept of the absolute negative.4 These generalizations of the number system and the consequent freedom of arithmetic from geometrical representation were to be essential in the development of the concepts of the calculus, but the Hindus could hardly have appreciated the theoretical significance of the change. Similarly, another consequence of the lack of nice distinction in Hindu thought happens to have been responsible for a change which corresponds to the modern view. We have seen that the crisis of 1

Cf. Karpinski, The History of Arithmetic, p. 46. Datta and Singh, History of Hindu Mathematics, Part I, p. 7. 3 Lasswitz, Geschichte der Atomistik, I, 185. * Fine, The Number System of Algebra, p. 105. 1

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the incommensurable led to the abandonment by the Greeks of the attempt to associate numbers with all geometrical magnitudes. The Pythagorean problem of the application of areas was in general insoluble in terms of the conceptions of number and geometrical magnitude then current. The area of the circle could not literally be exhausted by applying rectilinear configurations to it, inasmuch as curvilinear magnitudes were fundamentally different. The area of a circle was not to be compared with that of a square, for area was not a numerical concept, and equality in general—as distinct from equivalence— meant congruence. The number * would have had no meaning in Greek mathematics. With the Hindus the view was different. They saw no essential unlikeness between rectilinear and curvilinear figures, for each could be measured in terms of numbers; arithmetic and mensuration, rather than geometry and considerations of congruence, were fundamental. The strong Greek distinction between the discreteness of number and the continuity of geometrical magnitude was not recognized, for it was superfluous to men who were not bothered by the paradoxes of Zeno or his dialectic. Questions concerning incommensurability, the infinitesimal, infinity, the process of exhaustion, and other inquiries leading toward the conceptions and methods of the calculus were neglected. Operational difficulties were felt in dealing with the number zero— difficulties which led Brahmagupta to regard zero as an infinitesimal quantity which ultimately reduces to nought; and which caused Bhaskara to say that the product of a number and zero is zero, but that the number must be retained as a multiple of zero if any further operations impend.6 Yet these difficulties do not appear to have been considered with the intention of resolving the logical questions, implicit in the use of indeterminate forms, which were later to puzzle the early users of the calculus. Questions of limits, although implied in their work, were not expressly stated.6 The emphasis which Hindu mathematicians placed on the numerical aspect of the subject, together with the use of the Hindu numerals and of the principle of positional notation (the latter having been employed also by the Babylonians) did, of course, make more easily possible the develop. ' Datta and Singh, op. ext., p. 242. • Sengupta, "History of the Infinitesimal Calculus in Ancient and Mediaeval India," p. 224.

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ment of algebra, and, subsequently, that of the algorithmic procedures of the calculus. However, the logical concepts of the derivative and the integral are just as easily defined in terms of Greek numeration as of our own, so that Hindu mathematics added no thought essential to the development of these ideas. The Hindu numerals reached Europe through the medium of the Arabic civilization. This was preeminently eclectic, so that in Arabian mathematics we find both Greek and Hindu elements. The Arabic reckoning is based on that of Hindus, and Arabic trigonometry is Indian also in its use of the sine and the arithmetic form, rather than of the Hipparchan chord and geometric representation. Arabic geometry, however, shows the influence of Euclid and Archimedes, and Arabic algebra indicates a return to Greek geometric demonstration and the Diophantine avoidance of negative numbers. 7 The general character of Arabic algebra, however, is somewhat different from that of Diophantus' Arithmetic, for it is rhetorical rather than syncopated and deals mostly with problems of practical life, rather than with the abstract properties of numbers. 8 The whole trend of Arabic mathematics was, like that of the Hindus, directed away from the speculations on the incommensurable, continuity, the indivisible, and infinity—ideas which in Greek geometry were leading toward the calculus. Furthermore, additions to the classic Greek works—such as the treatise we have by Alhazen (or Ibn al-Haitham) on the measurement by infinitesimals of the paraboloid and on the summations of the cubes and the fourth powers of the positive integers9—were slight, for Arabic thought lacked the interest which was necessary to pursue further such fecund ideas. To the Arabs, however, we owe the preservation and transmission to Europe of much of the Greek work which would otherwise have been lost. Christian Europe had, since the time of Pappus, added practically nothing to traditional mathematical theory, and was, in fact, largely unfamiliar with the ancient treatises until, in the twelfth century, Latin translations were made from Arabic, Hebrew, and Greek 7 Fine, Number System; cf. also Paul Tannery, Notions historiques, p. 333, and Karpinski, Robert of Chester's Latin Translation of the Algebra of Al-Khowarizmi, p. 21. • Gandz, "The Sources of al-Khowarizmi's Algebra," pp. 263-77. 9 See Ibn al-Haitham (Suter), "Die Abhandlung über die Ausmessung des Paraboloids." Cf. also Wieleitner, "Das Fortleben der archimedischen Infinitesmalmethoden bis zum Beginn der 17. Jahr., insbesondere ueber Schwerpunkt bestimmungen."

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manuscripts. Before this time the work of Euclid was known chiefly through the enunciation, largely without proofs, of selected propositions given in the early sixth century by Boethius, in his Geometry}* The work of Archimedes fared no better, for by the sixth century the only works of his generally known (and commented upon by Eutocius) were those on the sphere and cylinder, on the measurement of a circle, and on the equilibrium of planes (that is, on the law of the lever).11 When in the twelfth and thirteenth centuries the Greek works began to appear in Latin translations, they did not meet with an enthusiastic reception on the part of European scholars, interested as these men were in theology and metaphysics. The interest of Roger Bacon and the appearance of works such as those of Jordanus Nemorarius show that in the thirteenth century there was no apathetical lack of mathematical activity; but the knowledge displayed at this time indicates an inadequate familiarity with the classic works of Greek geometry12—a nescience which had led to the designation fuga miserorum13—for the fifth proposition of Euclid's Elements.1* The inadequate attention paid to Greek geometry during the later medieval period was paralleled by a similar lack of zeal for Greek and Arabic algebraic methods. The thirteenth century opened auspiciously with the appearance, in 1202, of the Liber abaci of Leonardo of Pisa. This, however, was not followed by a comparable work for almost 300 years—that is, until the appearance in 1494 of the Summa de arithmetica of Luca Pacioli. The dearth of advances in the mathematical tradition during this period has been made the basis for very severe strictures on the mathematical work done in this interval.1® Such condemnation is justified by thinking in terms of contri10

Ball, A Short Account of the History of Mathematics, p. 107. See The Works of Archimedes, p. xxxv. Ginsburg, "Duhem and Jordanus Nemorarius," p. 361. u Later pons asinorum. 14 Smith, "The Place of Roger Bacon in the History of Mathematics," pp. 162-67. u Hankel (Zur Geschichte der Mathematik im Alterthum und Mittelalter, p. 349) says: "Mit Erstaunen nimmt man wahr, dass das Pfund, welches einst Leonardo der lateinischen Welt übergeben, in diesen drei Jahrhunderten durchaus keine Zinsen getragen hatte; wir finden, von Kleinigkeiten abgesehen, keinen Gedanken, keine Methode, welche nicht aus dem liber abaci oder der practica geometriae bereits wohl bekannt oder ohne Weiteres abzuleiten wäre." See also pp. 357-58 for a similar complaint. Cajori (A History of Mathematics, p. 125) says that in this period "the only noticeable advance is a simplification of numerical operations and a more extended application of them." Similarly, Archibald (Outline of the History of Mathematics, p. 27) characterizes the interval from u u

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butions to Greek geometry" and to Arabic algebra (to which, however, Nicole Oresme did add in the fourteenth century the conception of a fractional power17 which was later to add to the facility of application of the methods of the calculus). During the interval from 1202 to 1494 there may, indeed, have appeared no successor to either Archimedes or to Leonardo of Pisa.18 If, on the other hand, we regard the broader aspects of mathematics—the speculations and investigations which lead up to the propositions which are in the end deductively demonstrated—it will appear that this so-called barren period furnished points of view of significance in the development of the calculus. In this respect, as in others, there was perhaps as much originality in medieval times as there is now.1* Throughout the early period of the Middle Ages, Aristotle had been known in Europe largely through his logical works. During the thirteenth century, however, his scientific treatises circulated freely and, although these were condemned at Paris in 1210,20 their study was again established in the university by 1255, at which time nearly all of Aristotle was prescribed for candidates for the master's degree.21 In the Physica, Aristotle had considered in some detail the infinite, the infinitesimal, continuity, and other topics related to mathematical analysis. These became, particularly in the next century, the center of a lively discussion on the part of scholastic philosophers. They were studied in the light of Peripatetic philosophy, rather than in terms of mathematical postulational thought, but the resulting speculations were of service in sustaining an interest in such conceptions until, at a later date, they became a part of mathematics. Leonardo of Pisa to Regiomontanus as "a period of about 250 barren years." Bjömbo ("Über ein bibliographisches Repertorium der handschriftlichen mathematischen Literatur des Mittelalters," p. 326) says "Von einer Entwickelung in dieser Epoche ist kaum zu reden; bedeutende mathematische Fortschritte wird man hier vergebens suchen." " It has been well said by Sarton {Introduction to the History of Science, I, 19-20) that we do not do justice to the medievals because we judge their first steps by the Greek last steps. 17 Cf. Algorismus proportionum, pp. 9-10. Fine (Number System, p. 113) goes so far as to say that this is the only contribution of the period to algebra. u Eneström, "Zwei mathematische Schulen im christlichen Mittelalter." " See Sarton, op. cit., I, 16. 10 Denifle and Chatelain, Chartularium Universitaiis Parisiensis, Vol. I, p. 70, No. 11. Cf. also Vol. I, pp. 78-79, No. 20. 51 Ibid., Vol. I, pp. 277-79, No. 246. Cf. also Rashdall, The Universities of Europe in the Middle Ages, I, 357-58.

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One of the best examples of Scholastic thought on these topics is that found in the work of a man who exerted a great influence upon medieval thought,22 Thomas Bradwardine, "doctor profundus" and Archbishop of Canterbury, and perhaps the greatest English mathematician of the fourteenth century. In his Geometría speculative and in the Tractatus de continuo24 Bradwardine discussed, among other things, the nature of continuous magnitude, his view being dominated by the Peripatetic opposition to any atomism. The doctrine of Leucippus and Democritus, which had denied divisibility to infinity, has at all times had partisans and adversaries, and the Scholastic period was far from exceptional in this respect. The idea of the indivisible, during the earlier Middle Ages, was often more elementary than that held by Democritus long before. It seems to have been believed by Capella, Isidore of Seville, Bede, and others, that time is composed of indivisibles, an hour being made up of 22,560 such instants." It seems probable, or at least possible, that these instants were regarded as atoms of time. During the later medieval period the idea of indivisibles under various forms and modifications was upheld by Robert Grosseteste, Walter Burley, and Henry Goethals, among others.24 On the other hand, Roger Bacon protested, in his Opus majus, that the doctrine of indivisibles was inconsistent with that of incommensurability,27 an argument developed further by Duns Scotus, William of Occam, Albert of Saxony, Gregory of Rimini, and others.28 Bradwardine considered, in the light of the problem of the continuum, the divers points of view represented by proponents of the doctrine of indivisibles. Some interpreted the question in terms of physical atomism, others of mathematical points; some assumed a finite, others an infinite, number of points; some postulated immediate contiguity, others a discrete set of indivisibles.29 Bradwardine himself maintained a

Duhem, Les Origines de la statique, II, 323. ° For brief indications of the contents of this work, see Moritz Cantor, Vorlesungen, II, 103-6; and Hoppe, "Zur Geschichte der Infinitesimalrechnung," pp. 158-60. M For an analysis of this see Stamm, "Tractatus de continuo von Thomas Bradwardina" ; see also Cantor, Vorlesungen, II, 107-9. " Paul Tannery, "Sur la division du temps en instants au moyen âge," p. 111. M Stamm, "Tractatus de continuo," pp. 16-17; cf. also Duhem, Études sur Léonard de Vinci, II, 10-18. " Smith, The Place of Roger Bacon in the History of Mathematics, p. 180, n. * Duhem, Études sur Léonard de Vinci, II, 8. " Stamm, "Tractatus de continuo," p. 16.

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that continuous magnitudes, although including an infinite number of indivisibles, are not made up of such atoms.10 "Nullum continuum ex indivisibilibus infinitis integran vel componi," said Bradwardine,31 using perhaps for the first time in this connection the word which Leibniz was to adopt (upon the suggestion of the Bernoulli brothers*1) to designate in his calculus the sum of an infinite number of infinitesimals—the integral. Bradwardine asserted, on the contrary, that a continuous magnitude is composed of an infinite number of continua of the same kind. The infinitesimal, therefore, evidently possessed for him, as for Aristotle, only potential existence.33 William of Occam seems to have occupied a position intermediate between that of Bradwardine and that held by the supporters of indivisible Unes. While admitting that no part of any continuum is indivisible, he maintained that, contrary to the teachings of Aristotle, the straight line does actually (not only potentially) consist of points.34 In another connection, however, Occam said that points, lines, and surfaces are pure negations, having no reality in the sense that a solid is real.35 One must not, moreover, read too much into the Scholastic views on the nature of the continuum. The opinion of Bradwardine has been rather freely identified with that of Brouwer and the modern intuitionists, who conceive of the continuum as made up of an infinite number of infinitely divided continua;38 the view of Occam has been said to correspond to that held by Russell and the formalists, who regard the continuum as a perfect set of points everywhere dense.37 Such comparisons are justifiable only in a very general sense, for the Scholastic speculations invariably centered upon the metaphysical question of the reality of indivisibles, rather than upon the search for a representation which should be consistent with the premises of mathematics. There was in the medieval views no conception of the rigorous axiomatic foundation of arithmetic which has » Moritz Cantor, Vorlesungen, II, 108. "Moritz Cantor, Vorlesungen, II, 109, n.; Hoppe, "Zur Geschichte der InfinitesimaJrechnung," p. 159; cf. also Stamm, op. cit., p. 17. " James Bernoulli, "Analysis problematis antehac propositi," p. 218. 13 Lasswitz, Geschichte der Atomistik, II, 201; Stamm, "Tractatus de continuo," p. 17. M Bums, "William of Ockham on Continuity." 35 Duhem, Études sur Léonard de Vinci, II, 16-17; i n , 26. " Stamm, "Tractatus de continuo," p. 20. " See Birch, "The Theory of Continuity of William of Ockham," p. 496.

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characterized modern thought upon this subject. Nevertheless, the fourteenth century disputations on the indivisible represent a keen appreciation of the difficulties involved and a clarity of thought which was, several centuries later, to lend an air of respectability to the infinitesimal methods leading to the calculus. In discussions of indivisibles, the question of infinite division and the nature of the infinite arose, of necessity. In fact, the medieval philosophers discussed the question more from the point of view of infinite divisibility and infinite aggregates than from that of infinitely great magnitudes. Aristotle, it will be recalled, had distinguished two kinds of infinity—the potential and the actual. The existence of the latter he had categorically denied, and the former he had admitted as realized only in cases of infinitely small continuous magnitudes and of infinitely large numbers.58 The Roman poet Lucretius had, with keen imagination, upheld the notion of the infinite as indicating more than the potentiality of indefinite increase. In focusing attention upon infinite multitudes rather than magnitude, he adumbrated a number of properties of infinite aggregates, such as that a part may in this case be equal to the whole." The work of Lucretius was not, however, familiar to the European scholars of the Middle Ages. The Aristotelian distinction, on the other hand, was continued by the Scholastic philosophers, although with modifications resulting, perhaps, from the fact that Christianity recognized an infinite God. In the thirteenth century Petrus Hispanus, who became Pope John X X I , recognized in his Summidae logicales40 two kinds of infinity: a categorematic infinity, in which all terms are actually realized, and a syncategorematic infinity, which is bound up always with potentiality.41 This distinction is not greatly different from that which has been suggested recently by a mathematician and scientist who would discriminate between saying that an infinite aggregate is conceivable and saying it is actually conceived.42 The discussion of the two infinities, which was begun in the thirteenth century, was continued throughout the fourteenth also. Albert » Pkysica, Book III. *• See Keyser, "The Rôle of the Concept of Infinity in the Work of Lucretius." *> Duhem, Études sur Léonard de Vinci, H, 22. u Duhem, op. cit., Vol. II, Léonard de Vinci et les deux infinis, pp. 1-53, gives an extensive account of this question. " Enriques, Problems oj Science, pp. 127-28.

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of Saxony, for example, brought out the distinction nicely by a mere transposition of words, saying that the two views were illustrated respectively by the sentences: "In infinitum continuum est divisible," and "Continuum est divisible in infinitum."48 Bradwardine brought out the difference, perhaps with less subtlety but certainly more clearly, in saying that the categorematic infinity is a quantity without end, whereas the syncategorematic infinity is a quantity which is not so great but that it can be made greater." Although the distinction between the two infinities was generally recognized by the philosophers of the Scholastic period, there were significant differences, then as now, on the question of their existence. William of Occam, in conformity with his nominalistic attitude and with the principle of economy enunciated in his well-known "razor," agreed with Aristotle in denying that the categorematic infinity is ever realized. Gregory of Rimini, on the other hand, maintained4* what mathematics in the nineteenth century was to demonstrate: that there is in thought no self-contradiction involved in the idea of an actual infinity—the so-called completed infinite. More interesting from the mathematical point of view than these philosophical and discursive discussions are the remarks made on the subject of the infinite by Richard Suiseth, popularly known as the Calculator, in his Liber calculationum. This was composed later than 1328, inasmuch as it refers to Bradwardine's44 Liber de proportionibus 47 of that year, and probably dates from the second quarter of the fourteenth century.48 Although more interested in dialectical argu« Duhem, Études sur Léonard de Vinci, H, 23. Stamm, "Tractatus de continuo," pp. 19-20. « See Duhem, Études sur Léonard de Vinci, II, 399-401. 44 Duhem (Études sur Léonard de Vinci, HI, 429) has incorrectly asserted that Calculator names in his work only Bradwardine, Aristotle, and Averroes, but there are in the Liber calculationum references also to ancient mathematicians, for example, Euclid (fol. 29', cols. 1-2) and Boethius (fol. 43», col. 1). 41 "Ut venerabilis magister Thomas de berduerdino in suo libro de proportionibus liquide declarat." Liber calculationum, fol. 3*, col. 1. Professor Lynn Thomdike has kindly allowed me to make use of his rotograph of a copy in the British Museum, I B. 29, 968, fol. lr—83.® This is a copy of the undated editio princeps at Padua, placed by Thomdike (A History of Magic and Experimental Science, III, 372) in the year 1477. Inasmuch as this work is neither well known nor readily available, passages from it will be cited at some length. The printed text contains so many abbreviations that it is difficult to read, but transcriptions from it will be given in full. « See Thomdike, History of Magic, III, 375. Stamm ("Tractatus de continuo," p. 24) would place its appearance after 1350. 44

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ments and subtle sophisms concerning infinity than in its adequate definition, Suiseth made several comments on the subject which are of particular mathematical significance. In the second chapter he said that all sophisms regarding the infinite could be easily resolved by recognizing that a finite part can have no ratio to an infinite whole." This conclusion, he said, would be conceded by the imagination, for the contrary would imply that any part, when added to the whole, would not change it in magnitude. Arguments with respect to the infinite do not proceed, therefore, as do those concerning finite quantities. 50 About two hundred years later Galileo remarked still more clearly the essential difference between the rules for the finite and those for the infinite, but he focused attention upon the correspondence between infinite aggregates, rather than upon the ratio of finite to infinite magnitudes—a change of view which led to the final formulation of the calculus in the nineteenth century. There is in the statements of Calculator a cogency and a warning which might well have been observed in later centuries, but they display the Peripatetic propensity to regard the infinite as a magnitude, rather than as an aggregation of terms. The fruitfulness of the infinite in the work of Archimedes arose out of the infinite collections of lines or of terms in a series. Reference will be made below, to be sure, to the study by Calculator of an infinite series, but in this connection it will be seen that it was not the endlessness of the sequence of terms which most interested him, but a certain infinite magnitude. A consideration of this series will require, however, the study of a larger problem with which it was associated, and to which we shall now turn. The blending of theological, philosophical, mathematical, and scientific considerations which has so far been evident in Scholastic thought is seen to even better advantage in a study of what was per• "Infinita quasi sophismata possunt fieri de infinito que omnia si diligenter inspexeris quod nullius partis ad totum infinitum est aliqua proportio faciliter dissoluere poteris per predicta." /.tier calculaiionum, fol. 8", col. 2. " " Q u e potest concedi de inmaginatione et causa est quia nulla pars finita finite intensa respectu tocius infiniti aliquid confert quia nullam habet proportionem a d illud infinitum si tamen subiectum esset finitum conclusio non foret inmaginabilis quia tunc conclusio inmediate repugnaret illi positioni quia tunc quelibet pars in comparatione ad totum conferet aliquantum et sic non est nunc ideo nullum argumentum proceditur de infinito sicut faceret de finite." Ibid., fol. 8 r , col. 2, fol. 8", col. 1.

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haps the most significant contribution of the fourteenth century to the development of mathematical physics. It has commonly been protested that additions, if any, which were made to scientific knowledge in the medieval period lay solely in the field of practical discoveries and applications; and that the only mathematical achievement during this time was the simplification of the rules of operation for the Hindu-Arabic numerals, the latter having been made known in Europe by Leonardo of Pisa and other men of the thirteenth century. There is at least one exception to such an assertion, for it was precisely during this interval, and particularly in the fourteenth century, that a theoretical advance was made which was destined to be remarkably fruitful in both science and mathematics, and to lead in the end to the concept of the derivative. This consisted in the idea— often expressed, to be sure, in terms of dialectical rather than mathematical method—of studying change quantitatively, and thus admitting into mathematics the concept of variation.61 Heraclitus, Democritus, and Aristotle had made some qualitative metaphysical speculations on the subject of motion, and occasionally Greek geometers (Hippias, Archimedes, Nicomedes, Diocles) had allowed this notion to enter their thoughts (though not their proofs); but the idea of representing continuous variation by means of geometrical magnitude or of studying it in terms of the discreteness of number does not seem to have arisen with them. The Greek sciences of astronomy, optics, and statics had all been elaborated geometrically, but there was no such representation of the phenomena of change. Archimedes' famous work in statics was not paralleled by any equivalent kinematic system which admitted of representation in the form of mathematical propositions. Perhaps the demand for rigor in Greek thought, which made congruence fundamental in geometry and which allowed no confusion of M It is preposterous to assert—as does Tobias Dantzig {Aspects of Science, p. 45)— that, with reference to Aristotle and the Schoolmen, "his casuistry, his predilection for the static, his aversion for everything that moved, changed, flowed, or evolved admirably suited their purposes." A brief examination of the works of Aristotle on physical science and a glance at A Catalogue of Incipits of Mediaeval Scientific Writings in Latin by Thorndike and Kibre will show how untrue is such a statement. Aristotle made extensive investigations, from the point of view of physics, into the phenomena of motion. Scholastic philosophers not only continued his work, but also added the quantitative form of expression which was so successfully developed later in the seventeenth century.

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the discrete with the continuous, could not be met by any attempt to establish a science of dynamics. Moreover, Greek astronomy lacked the concept of acceleration; the motions involved were all uniform (and hence eternal) and circular, and could, in this case, be represented by the geometry of the circle. No such uniformity was apparent for local motion—that is, for terrestrial changes of position. Motion was, it appeared, a quality rather than a quantity; and there was among the ancients no systematic quantitative study of such qualities.52 Aristotle had spoken of mathematics as concerned with "things which do not involve motion," and had held that mathematics studies objects qua continuous, physics qua moving, and philosophy qua being.6* In general Greek mathematics was the study of form, rather than variability. The quantities entering into Diophantine algebraic equations are constants, rather than variables, and this is true also of Hindu and Arabic algebra. In the Scholastic period, however, there arose a problem which was ultimately to change this view. Aristotle had considered motion a quality that does not increase and decrease through the joining together of parts, as does a quantity," and this idea dominated most thought until toward the end of the thirteenth century," at which time a reaction against Peripateticism arose, which was to lead, at Paris, to new views on the subject of motion." Bacon, in the second half of the thirteenth century, still followed the Aristotelian discussion of motion,57 but a new approach to the problem was evidenced early in the fourteenth century by the introduction of the idea of impetus—the notion that a body, once set in motion, will continue to move because of an internal tendency which it then possesses, rather than, as Peripatetic doctrine had taught, because of the application of some external force, such as that of the air, which continues to impel it. This doctrine, which has been ascribed to Jean Buridan,58 was particularly significant as an adumbration of the famous work in dynamics of Galileo almost three centuries later. At " Brunschvicg, Les Étapes de la philosophie mathématique, p. 97. B Physica H. 198a; Metapkysica 1061. M Duhem, Études sur Léonard de Vinci, III, 314-16; cf. also Wieleitner, "Ueber den Funktionsbegriff und die graphische Darstellung bei Oresme," pp. 196-97. " Wieleitner, "Ueber den Funktionsbegriff," p. 197. M Duhem, Études sur Léonard de Vinci, H, pp. iii-iv. n Thomson, "An Unnoticed Treatise of Roger Bacon on Time and Motion." u See Duhem, Études sur Léonard de Vinci, passim.

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the time of its inception it served also to make more acceptable the intuitive notion of instantaneous velocity, an idea excluded by Aristotle from his science, but implied by the quantitative study of variation of the fourteenth century. At that stage no precise definition of instantaneous rate of change could, of course, be given—nor was one given by Galileo—but there appeared at the time a large number of works, more philosophical than mathematical, all based on the intuition of this concept which everyone thinks he possesses. These tractates were devoted to a discussion of the latitude of forms, that is, of the variability of qualities. There seems to be no scientific term which correctly expresses the equivalent of the word form as here used. It refers in general to any quality which admits of variation and which involves the intuitive idea of intensity—that is, to such notions as velocity, acceleration, density. These concepts are now expressed quantitatively in terms of limits of ratios—that is, simply as numbers—so that no need is now felt for a word to express the medieval idea of a form. In general, the latitude of a form was the degree to which the latter possessed a certain quality, and the discussion centered about the intensio and the retnissio of the form, or the alterations by which this quality is acquired or lost. Aristotle had distinguished between uniform and nonuniform velocity, but the critical dialectical treatment of the Scholastics went much further. In the first place, the time rates of change which they considered were not necessarily those of distance, but included many others as well, such as those of intensity of illumination, of thermal content, of density. Secondly—and more significantly—they distinguished not only between latitudo uniformis and latitudo dijformis (that is, between uniform and nonuniform rates of change), but proceeded further to classify the latter as either latitudo uniformiter dijformis or latitudo dijformiter dijformis (that is, according as the instantaneous rate of change of the rate of change was uniform or not); and the last-mentioned were sometimes in turn further divided into either latitudo uniformiter dijformiter dijformis or latitudo difformiter dijformiter dijformis. These attempts to introduce order, by means of verbal arguments, into the disconcerting problem of variability were destined to be replaced centuries later by equivalent statements, expressed in mathe-

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matics and science by the remarkably concise terminology and symbolism of algebra and the differential calculus. At the time at which they were made, however, they represented the first serious and careful effort to make quantitative the idea of variability. The origin of the question of the latitude of forms is shrouded in doubt. Duns Scotus appears to have been among the first to consider the increase and the decrease (intensio and remissio) of forms,6' although the loose idea of latitude of forms apparently goes back to some time before this, inasmuch as Henry Goethals used the word latitude in this connection.60 In the early part of the fourteenth century there appeared works on variability and the latitude of forms by James of Forli, Walter Burley, and Albert of Saxony.61 Similar ideas appeared also in 1328, at Oxford, in the treatise on proportions of Bradwardine.62 This work is devoted more particularly to mechanics than to arithmetic, but the archbishop did not make a special investigation into the theory of the latitude of forms.63 "In the fourteenth century the study of the mathematical sciences flourished greatly in Oxford,"64 and it was here that appeared not only the work of Bradwardine, but also the "leading model"66 of such treatises on the latitude of forms—the Liber calculationum of Suiseth to which reference has already been made.66 That the doctrine of the latitude of forms was well known by the middle of the fourteenth century is indicated by the fact that Calculator begins in médias res by a general consideration, in the first chapter, of the intension and remission of forms and the question as to whether a latitudo dijjormis corresponds to its maximum or minimum gradus or intensity. The sense in which a form could correspond to any gradus is not clear, but Calculator appears here to be striving «• Von Prantl, Gesckichte der Logik im Abendlande, III, 222-23. Cf. Stamm, "Tractatus de continuo," p. 23. M Duhem, Eludes sur Léonard de Vinci, III, 314-42. " Wieleitner, "Der 'Tractatus de latitudinibus formarum des Oresme,' " pp. 123-26. " Duhem, Études sur Léonard de Vinci, III, 290-301. M Stamm, "Tractatus de continuo," p. 24. M Gunther, Early Science in Oxford, II, 10. « Thoradike, History of Magic, n i , 370. •• For a general description of the contents of this work, see Thorndike, History of Magic, Vol. n i , Chap. XXIII; and Duhem, Études sur Léonard de Vinci, m , 477-81. No analysis of the Liber calculationum from the mathematical point of view appears to be available.

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toward the idea of average intensity, an idea which could not be made precise without the use of the concepts of the calculus. In Chapter II, however, he arrived, in this connection, at a result which was to be of particular significance in the later development of science and mathematics. Here, in considering the intension in difform things, he reached the conclusion, in connection with problems on thermal content, that the average intensity of a form whose rate of change over an interval is constant, or of a form which is such that it is uniform throughout each half of the interval, is the mean of its first and last intensities." The rigorous proof of this requires the use of the limit concept, but Calculator had resort to dialectical reasoning, based on physical experience of rate of change. He argued, in connection with such a form, that if the greater intensity is allowed to decrease uniformly to the mean while the lesser is increased at the same rate to this mean, then the whole is neither increased nor decreased.68 The argument is pursued at further length here, and the author adds, as well, numerical illustrations such as the following: If the intensity increases uniformly from four to eight, or if for the first half of the time it is four, and for the last half it is eight, then the effect is that which would result from a uniform intensity of six, operating throughout the whole time. 6 ' The methods presented here are amplified and applied, in other chapters of the Liber calculationum, to questions dealing with density, velocity, and the intensity of illumination. In these Suiseth again made use of the law that if the rate of change of one of these is constant, or if the form is such that in each of the two halves of the " "Primo arguitur latitudinem caliditatis uniformiter difformem seu etiam difformem cuius utraque medietas est unifonms, et calidum uniformiter difforme suo gradui medio correspondere." Liber calculationum, fol. 4*, col. 2. u "Capiatur talis caliditas seu tale calidum et remittatur una medietas ad medium et intendatur alia ad medium equeuelociter: et sequitur totum non intendi nec remitti; eo quod totam latitudinem acquiret. Secundum unam medietatem seu partem sicut deperdet secundum aliam partem equalem et in fine erit uniforme sub tali gradu medio. Igitur nunc correspondet tali gradui." Ibid. •• "Sit enim tale uniformiter difforme seu difforme cuius utraque medietas est uniformis una ut. VIII. et alia ut J U L gratia argumenti. Tunc prima qualitas ut .VIII. extenditur per medietatem totius per predicta. Ergo solum denominat totum ut quatuor per idem prima qualitas ut quatuor per aliam medietatem extensa solum facit ut duo ad totius denominatorem. Igitur ille due qualitates totum precise denominabunt ut .VI. qui est gradus medius inter illas medietates. Sequitur igitur positio sic in speciali." Ibid., fol. 5*, col. 2.

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interval the rate of change is zero, then the average intensity is the mean of the first and last values. As was the case in the study of variation of the intensity of heat, no definition is given of the terms employed, inasmuch as this would presuppose an appreciation of the limit concept. The lack of such precise definitions led Calculator into difficulties involving the infinite. He considered, for example, a rarity of degree zero as a density of infinite degree, and conversely,70 and consequently became involved unnecessarily often in the paradoxes of the infinite. The tendency displayed by Calculator to consider the infinite from the point of view of intensity or magnitude, rather than aggregation, is brought out again in the discussion of an example of nonuniform variation, which might have had a significant influence upon the development of the calculus, had Suiseth's purpose been less that of introducing mathematics into dialectical discussions of change and more that of bringing the problem of variation into the realm of mathematics. Calculator, in the second book of his Liber calculationum, had occasion to consider the following problem: if throughout half of a given time interval a variation continues at a certain intensity, throughout the next quarter of the interval at double this intensity, throughout the following eighth at triple this, and so ad infinitum; then the average intensity for the whole interval will be the intensity of the variation during the second subinterval (or double the initial intensity). 71 This is equivalent to a summation of the infinite series 2 + 1 + n 1 + A + . . . + — + . . . = 2. I t will be recalled that Archimedes 2"

"Ergo .a. est infinite densum et per consequens non est rarum. Ex istis sequitur ista conclusio quod aliquid est rarum quod non est rarum quia .a. est rarum quia est uniformiter difforme rarum et non est rarum quia est infinite densum. Pro istis negatur utraque conclusio et tunc ad casum positum quod .a. sit unum uniformiter difformiter rarum terminatur ad non gradum raritatis negatur casus nam ex quo raritas se habet privative sequitur ut argutum est quod ab omni gradu raritatis usque ad non gradum raritatis est latitudo infinita quia non gradus raritatis est infmitus gradus densitatis sed impossible est quod aliquid sit uniformiter difforme aliquale mediante latitudine infinita. Ideo casus est impossibilis sicut est impossibile quod aliquid sit uniformiter difforme remissum ad non gradum remissionis terminatum." Ibid., fol. 19*, cols. 1-2. 71 "Contra quam positionem et eius fundamentum arguitur sic quia sequitur quod si prima pars proportionalis alicuius esset aliqualiter intensa, et secunda in duplo intensior, et tertia in triplo intensior, et sic in infinitum totum esset equate intensum precise sicut est secunda pars proportionalis quod tamen non uidetur visum." Ibid., fol. 5 , col. 2. 70

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had made use of certain simple series in connection with his geometry» but that the characteristic Greek interdiction of the infinite had led him to consider the sum to n terms only. It was apparent to him that the series 1 + 1 + t t + t t + . . . + approached i in such a way that the difference could be made, by taking a sufficiently large number of terms, less than any specified quantity. He did not, however, go so far as to define I as the "sum" of the infinite series, for this would have exposed his thought to the paradoxes of Zeno, unless he had invoked the precisely formulated concept of a limit as given in the nineteenth century. The Scholastic discussions of the fourteenth century, on the other hand, referred frequently to the infinite, both as actuality and as potentiality, with the result that Suiseth, with perfect confidence, invoked an infinite subdivision of the time interval to obtain the equivalent of an infinite series. He did not resolve the aporias of Zeno, to show in what sense an infinite series may be said to have a sum—a problem which future mathematicians were to consider at length. Calculator, instead, was more particularly interested in infinite magnitudes than in infinite series. Not only is the time interval in his problem infinitely divided, but the intensity itself becomes infinite. Now how can a quantity, whose rate of change becomes infinite, have a finite average rate of change? Suiseth admitted that this paradoxical result was in need of demonstration and so furnished at great length the equivalent of a proof of the convergence of the infinite series. This he did as follows. Consider two uniform and equal rates of change, a and b, operating throughout a given time interval, which has been subdivided in the ratios 3, j, I, . . . . Now let the rate of change b be doubled throughout the interval; but in the case of a, let it be doubled in the second subinterval; tripled in the third; and so to infinity, as given in the problem above. Now the increase in a in the second subinterval, if continued constantly throughout this and all following subintervals, would result in an increase in the effect equal to that brought about by the change in b during the first half of the time. The tripling of a in the third subinterval, if continued constantly throughout this and the ensuing subintervals, would in turn result in a further increase in the effect of a equal to that brought about by the change in b in the second subinterval, and so to infinity. Hence the increase resulting

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from the doubling, tripling, and so forth of a is equal to that caused by the doubling of b; i. e., the average rate of change in the problem considered above is the rate of change during the second subinterval, which was to be proved. 72 The tediously verbose proof given by Calculator is, of course, based entirely on arguments appealing to our intuition of uniform rate of change. Because Suiseth gave no unambiguously clear definitions of velocity, density, intensity of illumination, and other terms used freely in his dissertation, his work not infrequently presents—as had also, to a certain extent, the Physica of Aristotle 73 —the appearance of an effort to propound sophistical questions on the subject of change, rather than that of a serious effort to establish a scientific basis for the study of the phenomena of motion and variability. 74 n "Nam apparet quod ilia qualitas est infinita ergo si sit sine contrario infinite denominabit suum subiectum. Et quod conclusio sequatur arguitur sic: sint .a. .b. duo uniforma eodem gradu, et dividatur .b. in partes proportionales et est ilia hora ita quod partes maiores terminentur seu incipiant ab hoc instanti, et ponatur quod in prima parte proportionali illius hore intendatur prima pars .b. ad duplum, et in secunda parte proportionali intendatur secunda pars proportionalis illius ad duplum, et sic in infinitum, ita quod in fine erit .b. uniforme sub gradu duplo ad gradum nunc habitum. E t ponatur quod .a. in prima parte proportionali illius hore intendatur totum residuum .a. prima parte proportionali .a. acquirendo totam latitudinem sicut tunc acquireret prima pars proportionalis ,b. et in secunda parte proportionali euisdem hore intendatur totum residuum .a. a prima parte proportionali et secunda illius .a. acquirendo tantam latitudinem sicut tunc acquiret pars proportionalis secunda .b. et in tertia parte proportionali indendatur residuum a prima parte proportionali et secunda et tertia acquirendo tantam latitudinem sicut tunc acquiret tertia pars proportionalis .b. et sic in infinitum scilicet quod quandocumque aliqua pars proportionalis .b. intendetur pro tunc intendatur .a. secundum partes proportionales subséquentes partem correspondentem in .a. acquirendo tantam latitudinem sicut acquiret pars prima in .b. et sint .a.b. consimilia quantitatiue continue quo posito sequitur quod .a. et .b. continue equeuelociter intendentur quia .a. continue per partes proportionales similiter intendetur sicut .b. quia residuum a prima parte proportionali .a. est equale prime parti proportionali eidem. Cum igitur .b. in prima parte proportionali illius hore continue intendetur per primam partem proportionalem et .a. per totum residuum a prima sua parte proportionali patet quod .a. in prima parte proportionali equeuelociter intendetur cum .b. et sic de omni alia parte eo quod quandocumque .b. indendetur per aliquam partem proportionalem .a. intendetur per totum interceptum inter partes correspondentes sibi et extremum ubi partes terminantur scilicet minores. Cum ergo quelibet pars proportionalis cuiuslibet continui sit equalis toti intercepto inter eandem et extremum ubi partes minores terminantur. Igitur patet quod .a. continue equeuelociter intendetur cum .b. et nunc est eque intensum cum .b. ut ponitur in casu. Ergo in fine .a. erit .a. eque intensum cum .b. et .a. tunc est tale cuius prima pars proportionalis erit aliqualiter intensa, et secunda pars proportionalis in duplo intensior, et tertia in triplo intensior et sic in infinitum. E t .b. erit uniforme gradu sub quo erit secunda pars proportionalis .a. ergo sequitur conclusio." Ibid., fol. 5', col. 2—fol. ( f , col. 1.

" Cf. Mach, The Science of Mechanics, p. 511. u "Multa alia possent fieri sophismata per rarefactionem subiecti, et per fluxum qualitatis et alterations qualitatis si subiectum debet intendi et remitti per huiusmodi rare-

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Suiseth's unfortunate adoption of the Peripatetic attitude that such qualities as dryness, coldness, and rarity are the opposites of moistness, warmth, density—rather than simply degrees of the latter7®— complicated his consideration of these through the unnecessarily frequent introduction of the paradoxes of the infinite. Nevertheless, to Calculator we owe perhaps the first serious effort to make quantitatively understandable these concepts of mathematical physics. His bold study of the change of such quantities anticipated not only the scientific elaboration of these, but also adumbrated the introduction into mathematics of the notions of variable quantity and derivative. In fact, the very words jiuxus and fluens, which Calculator used in this connection,76 were to be employed by Newton some three hundred years later, when in his calculus he spoke of such a variable mathematical quantity as a fluent and called its rate of change a fluxion. Newton apparently felt as little need as Suiseth for a definition of this notion of fluxion, and was satisfied to make a tacit appeal to our intuitions of motion. Our definitions of uniform and nonuniform rate of change, are, as Suiseth anticipated, numerically expressed; but their rigorous definition could be given only after the development, to which Newton contributed, of the limit concept. This latter arose out of the notions of the calculus, which, in their turn, had evolved from the intuitions of geometry. The prolix dialectic of Calculator made no appeal to the geometrical intuition, which was to act as an intermediary between his early attempts to study the problem of variation and the final formulation given by the calculus. This link between the interminable discursiveness of Suiseth and the concise symbolism of algebra was supplied by others of the fourteenth century who studied the latitude of forms. Of these the most famous was Oresme. Nicole Oresme was born about 1323 and died in 1382. He was thus somewhat younger than Calculator, from whom we have a manuscript (actionem, et fluxum et alterationem, ad que omnia considerando proportionem totius ad partem responsionem alicere poteris ex predictis faciliter." Liber calcidationum, fol. 91, col. 2. The word fluxum has been italicized by me for emphasis. " It should be remarked, in this connection, that several centuries later Galileo entered into an extensive discussion as to whether the Peripatetic qualities are to be regarded as positive (Le Opere di Galileo Galilei, I, 160 ff.), and that even in the eighteenth and nineteenth centuries the question as to the existence of "frigerific" rays was still argued. See, for example, Neave, "Joseph Black's Lectures on the Elements of Chemistry," p. 374. 78 Liber caiculationum, fol. 9', col. 2; fol. 75», col. 1.

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dated 1337, and his doctrines were in all probability derived from those of Suiseth and others of the Oxford School.78 Furthermore, from the very character of their work it would seem reasonable to place the composition of the Liber calculationum at Oxford before the appearance of the work of Oresme at Paris, on the grounds that the latter lacks the chief defects of the former—the prolixity of the complicated dialectical demonstrations and the author's propensity for subtle sophisms. There is in the entire Liber calculationum no diagram or reference to geometrical intuition, the reasoning being purely verbal and arithmetical.7' On the other hand, Oresme felt that the multiplicity of types of variation involved in the latitude of forms is discerned with difficulty, unless reference is made to geometrical figures.40 The work of Oresme therefore makes most effective use of geometrical diagrams and intuition, and of a coordinate system, to give his demonstrations a convincing simplicity. This graphical representation given by Oresme to the latitude of forms marked a step toward the development of the calculus, for although the logical bases of modern analysis have recently been divorced as far as possible from the intuitions of geometry, it was the study of geometrical problems and the attempt to express these in terms of number which suggested the derivative and the integral and made the elaboration of these concepts possible. The Tractatus de latitudinibus formarum81 has been generally 77

See Thomdike, History of Magic, III, 375. Duhem (Éludes sur Lionard de Vinci, I I I , 478-79) would see in the Liber calculationum the influence of Oresme and would characterize the work of Suiseth as "l'oeuvre d'une science sénile et qui commence á radoter." Wieleitner ("Zur Geschichte der unendlichen Reihen im christlichen Mittelalter," pp. 166-67) refers to Duhem in saying that Calculator rediscovered some of Oresme's examples. Thomdike has clearly shown, however, that Oresme's work was subsequent to that of Calculator. " Stamm ("Tractatus de continuo," p. 23), apparently referring to an edition which appeared in 1520 and which he mistakenly holds to be the first, asserts that Suiseth illustrated his representation with geometrical figures. Such illustrations may well have been later interpolations, inasmuch as none such appear in the first edition (1477). Furthermore, French copyists sometimes drew diagrams on the edge of manuscripts of such Oxford works (Duhem, Éludes sur Lionard de Vinci, III, 449). 10 "Quia formarum latitudines multipliciter variantur que multiplicitas difficiliter discemitur nisi ad figuras geométricas considerado referatur." Oresme, Tractatus de latitudinibus formarum, fol. 201r. I have made use of a photostat of a manuscript of this work in the Bayerische Staatsbibliothek, Miinchen, cod. lat. 26889, fol. 201";—fol. 206 f . See also Funkhouser, "Historical Development of the Graphical Representation of Statistical Data," p. 275; cf. also Stamm, op. cit., p. 24. n See Wieleitner, "Der 'Tractatus'," for a comprehensive description of this work. For comments on the work of Oresme and a study of this period as a whole, see Duhem, Études sur Lionard de Vinci, m , 346-405. n

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ascribed to Oresme and cited as his chief work on the subject. Although the views expressed in it are probably to be attributed to him, the treatise appears to be only a poor imitation, probably by a student,® of a larger work by Oresme entitled Tractatus de figuratione potentiarum et mensurarumThis latter work, one of the fullest on the latitude of forms, was written probably before 1361." It opens with the representation of variation by means of geometry, rather than with a dialectical exposition in terms of number, such as Calculator had given. Following the Greek tradition, which regarded number as discrete and geometrical magnitude as continuous, Oresme was led naturally to associate continuous change with a geometrical diagram. An intension, or rate of change by which a form acquires a quality, he imagined as represented by a straight line drawn perpendicular to a second line, the points of which represent the divisions of the time or space interval involved." For example, a horizontal line or longitude may represent the time or duration of a given velocity, and the vertical height or latitude the intensity of the velocity.8* It will be seen that the terms "longitude" and "latitude" are here used to represent, in a general sense, what we should now designate as the abscissa and the ordinate. The work of Oresme does not, of course, represent the earliest use of a coordinate system, for ancient Greek geography had employed this freely; nor can his graphical representation be regarded as equivalent to our analytic geometry,87 for it lacks the fundamental " Duhem, Éludes sur Léonard de Vinci, IH, 399-400. ° This work is known also by other titles, such as De unijormitate et difformitate intentionum, and De configuraiione qualitatum. See Wieleitner, "Ueber den Funktionsbegriff," for an extensive description of the portions of this work which are of significance in mathematics. For an analysis of those sections of the De configuraiione qualitatum which deal with magic, see Thomdike, History of Magic, Vol. I l l , Chap. XXVI. - Wieleitner, "Ueber den Funktionsbegriff," p. 198; Duhem, Études sur Léonard de Vinci, n i , 375. » "Omnis res mensurabilis extra numéros ymaginatur ad modum quantità tes continue. Ideo opportet pro ejus mensuratione ymaginare puncta, superficies et lineas aut istorum proprietatis, in quibus, ut voluit Aristoteles, mensura seu proportio perprius reperitur . . . . Omnis igitur intensio successive acquisibilis ymaginanda est per rectam perpendiculariter erectam super aliquod puctum aut aliquot puncta extensibilis spacii vel subjecti." Wieleitner, "Ueber den Funktionsbegriff," p. 200. 86 "Tempus itaque sive duratio erit ipsius velocitatis longitudo et ejusdem velocitatis intensio est sua Iatitudo." Ibid., pp. 225-26. See also the diagrams in Tractatus de latitudinibus formarum, fols. 202f and 205'. B Duhem has freely referred to Oresme as the inventor of analytic geometry. See, for example, his article on Oresme in the Catholic Encyclopedia; cf. also his Études sur Léonard de Vinci, HI, 386.

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notion that any geometric curve can be associated, through a coordinate system, with an algebraic equation, and conversely. 88 However, Oresme's work marks a notable advance in mathematical analysis in that it associated the study of variation with the representation by coordinates. Although Aristotle had denied the existence of an instantaneous velocity, the notion had continued to be invoked implicitly, upon occasion, by Greek geometers and by Scholastic philosophers. Oresme, however, was apparently the first to take the significant step of representing an instantaneous rate of change by a straight line. 8 ' He could not, of course, give a satisfactory definition of instantaneous velocity, but he strove to clarify this idea by remarking that the greater this velocity is, the greater would be the distance covered if the motion were to continue uniformly at this rate.' 0 Maclaurin, in attempting to clarify the Newtonian idea of a fluxion, expressed himself very similarly almost four hundred years later, but a rigorous and clear definition could be given only after the concept of the derivative had been developed. Oresme, furthermore, was confused by the perplexing problem of the indivisible and the continuum. In spite of his clear assertion that an instantaneous velocity is to be represented by a straight line, he accepted the dictum of Aristotle that every velocity persists throughout a time. 91 This view implies a form of mathematical atomism which, although underlying much of the thought leading to the calculus of Newton and Leibniz, has been rejected by modern mathematics. There is a widespread belief that the science of dynamics, which in the seventeenth century played such a significant role in the formation of the calculus, was almost entirely the product of the genius of Galileo, who "had to create . . . for us" 92 the "entirely new notion . . . of acceleration." 93 That such a view is a gross misconception will be clear to anyone who makes even a cursory examination of the fourteenth-century doctrine of the latitude of forms. Oresme, for » Cf. Coolidge, "The Origin of Analytic Geometry," p. 233. " "Sed punctualis velocitas instantanea est ymaginanda. per lineam rectam." Wieleitner, "Ueber den Funktionsbegriff," p. 226. " "Verbi gratia; gradus velocitatis descencus est major, quo subjectum mobile magis descendit vel descenderet si continuaretur simpliciter." Ibid., p. 224. n "Omnis velocitas tempore durat." Ibid., p. 225. ** M a c h , The Science of Mechanics, p . 133. See also Hogben, Science for the Citizen,

p. 241.

• M a c h , op. cit., p. 145.

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example, had a clear conception not only of acceleration in general but also of uniform acceleration in particular. This is evident from his statement that if the velocitatio (acceleration) is uniformis, then the velocitas (velocity) is unijormiter dijformis; but if the velocitatio is dijformis, then the velocitas is dijformiter dijformis. Oresme went further and applied his idea of uniform rate of change and of graphical representation to the proposition that the distance traversed by a body starting from rest and moving with uniform acceleration is the same as that which the body would traverse if it were to move for the same interval of time with a uniform velocity which is one-half the final velocity. Later, in the seventeenth century, this proposition played a central rôle in the development both of infinitesimal methods and of Galilean dynamics. It had been stated earlier and in a more c

FIGURE 6 general form by Calculator, but Oresme and Galileo gave to it a geometrical demonstration which, although not rigorous in the modern sense, was the best which could be furnished before the integral calculus had been established.94 The proof of the theorem is based upon the fact that the motion under uniform velocity, inasmuch as the latitude is the same at all times, is represented by a rectangle such as ABGF (fig. 6) and that the uniformly accelerated motion, in which the ratio of the change in latitude to the change in longitude is constant,96 corresponds to M

In the mistaken belief that the work of Oresme preceded that of Calculator, Duhem

(Éludes sur Léonard de Vinci, III, 388-98) has called this proposition the law of Oresme. M See Wieleitner, "Ueber den Funktionsbegriff," pp. 209-10; cf. also Tractatus de latitudinibits formarum, fol. 204 r . Duhem, upon the basis of these observations, has unwarrantedly asserted that Oresme "gives the equation of the right line, and thus forestalls Descartes in the invention of analytical geometry." See his article, "Oresme," in the Catholic Encyclopedia.

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the right triangle ABC. Oresme did not explicitly state the fact— this is, of course, demonstrated in the integral calculus—that the areas ABGF and ABC represent in each case the distance covered; but this seems to have been his interpretation," inasmuch as from the congruence of the triangles CFE and EBG he concluded the equality of the distances.'7 This is perhaps the first time that the area under a curve was regarded as representing a physical quantity, but such interpretations were to become before long commonplaces in the application of the calculus to scientific problems. Oresme did not explain why the area under a velocity-time curve represented the distance covered. It is probable, however, that he thought of the area as made up of a large number of vertical lines or indivisibles, each of which represented a velocity which continued for a very short time. Such an atomistic interpretation is in harmony with the views he expressed on instantaneous velocities and with the Scholastic interest in the infinitesimal. Several centuries later this interpretation was enunciated more boldly and vividly by Galileo, at a time when atomistic conceptions enjoyed an even greater popularity in both science and mathematics. Arguments similar to those given by Oresme and Calculator appeared also in the work of other men of the time, particularly at Oxford and Paris.'8 William of Hentisbery, a famous logician at Oxford who perhaps outdid Calculator in propounding sophisms on the subject of motion, clearly stated the law for uniformly difform variation." Marsilius of Inghen, at Paris, expounded this same law on the basis of Oresme's geometrical representation.100 The traditions developed at Oxford and Paris were continued also in the fifteenth century in Italy, where Blasius of Parma (or Biagio Pelicani), reputed the most versatile philosopher and mathematician of his time, explained the law similarly, in his Questiones super tractatum de latitudinibus formarum.im This same principle for the uniformly difform acquisition of qualities was known at Paris in the sixteenth century to Alvarus « Duhem, Études sur Líonard de Vinci, m , 394. " Wieleitner, "Ueber den Functionsbegriff," p. 230. " See Duhem, Études sur Líonard de Vinci, III, 405-81. " Wieleitner ("Der 'Tractatus'," pp. 130-32, n.) says that Hentisbury's arguments are accompanied by diagrams, but Duhem (Études sur Líonard de Vinci, III, 449) says there is no geometrical demonstration given with his writings. "»Duhem, Études sur Lionard de Vinci, m , 399-405. m See Amodeo, "Appunti su Biagio Pelicani da Parma."

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1

Thomas, John Major, Dominic Soto, and others. " The principles of uniformly accelerated motion thus seem to have been common knowledge to the Scholastics from the fourteenth century to the sixteenth, and it is very probable that Galileo was familiar with their work and made use of it in his development of dynamics.104 At all events, when the famous Discorsi of Galileo appeared in 1638, it contained a diagram and a type of argument resembling strikingly that previously given by Oresme.105 The efforts toward the mathematical representation of variation which resulted, in the fourteenth century, in the flurry of treatises on the latitude of forms, were tied up in many ways with other questions related to the calculus. While he was discussing, in connection with lalitudo dijfor miter difformis, a form represented graphically by a semicircle, the author of the Tractatus de latitudinibus formarum remarked that the rate of change of an intensity is least at the point corresponding to the maximum intensity.106 Great things have been read into this casual remark, for historians of mathematics have ascribed to its author the sentiments that the increment in the ordinate of the curve is zero at its maximum point, and that the differential coefficient vanishes at this point!107 Ascribing such ideas to the author of the Tractatus is obviously unwarranted, inasmuch as they presuppose the clear conceptions of limit and differential quotient which were not developed until many centuries later.108 Further1(B Wieleitner, "Zur Geschichte der unendlichen Reihen im christlichen Mittelalter," p. 154. "» Duhem, Études sur Léonard de Vinci, i n , 531 ff. Zeuthen, Geschichte der Mathematik im XVI. und XVII. Jahrhunderl, pp. 243-44. 1M Mach (The Science of Mechanics, p. 131), probably unaware of the earlier work, has ascribed the diagram and ideas to Galileo. 1M "In qualibet talis figura sua intensio terminatur ad summum gradum tarditatis et sua remissio incipit a summo gradu tarditatis ut in medio puncto aliqualis ubi terminatur intensio et incipit remissio, patet in figuris .c.d. et .d.c." Tractatus de latitudinibus formarum, fol. 205f. The figures referred to (fol. 204') are much like semicircles. m Curtze ("Ueber die Handschrift R. 4°, 2," p. 97) says that Oresme noticed in general that "der Zuwachs . . . . der Ordinate einer Kurve in der unmittelbaren Nähe eines Maximums oder minimums gleich Null ist," and Moritz Cantor makes an even stronger Statement when he says ( Vorlesungen, II, 120) "Oresme's Augen offenbarte sich die Wahrheit des Satzes, den man 300 Jahre später in die Worte kleidete, an den Höhen- und Tiefpunkten einer Curve sei der Differentialquotient der Ordinate nach der Abscisse Null." 109 Wieleitner ("Der 'Tractatus,' " p. 142) says "Ich halte aber die Bemerkung doch nur für rein anschauungsmässig"; and Timtchenko ("Sur un point du 'Tractatus de latitudinibus formarum' de Nicholas Oresme") agrees, saying "qu'il était encore assez loin du

théorème exprimé par la formule — = 0 (pour le sommet de la figure).''

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more, the writer had apparently no idea of generalizing the statement by extending the conclusion to other cases. His remarks do show clearly, however, how fruitful the idea of the latitude of forms was to be when it entered the geometry and algebra of later centuries, to become eventually the basis of the calculus. The Scholastic philosophers were striving to express their ideas in words and geometrical diagrams, and were not so successful as we who realize, and can make use of, the economy of thought which mathematical notation affords. difformis Further consideration by Oresme of a latitudo dijfor miter led him, as it had also Suiseth somewhat earlier, to another topic essential in the development of the calculus—that of infinite series. In the same manner as Calculator, he considered a body moving with uniform velocity for half a certain period of time, with double this velocity for the next quarter of the time, with three times this velocity for the next eighth, and so ad infinitum. He found in this case that the total distance covered would be four times that covered in the first half of the time. However, whereas Calculator had had recourse to devious verbal argumentation in his justification of this result, the method of Oresme was here (as in his earlier work) geometrical and consisted in the comparison of the areas corresponding to the distances involved in the graphical representation of the motion.109 In this manner also he handled another similar case in which the time was divided into parts J , A , ^r, . . . the velocity increasing in arithmetic proportion as before. The total distance he found this time to be that covered in the first subinterval. Oresme then went on to still more complicated cases. For example, he assumed that during the first half of the time interval the body moved with uniform velocity, then for the next quarter of the time with motion uniformly accelerated until the velocity was double the original, then for the next eighth of the time uniformly with this final velocity, then for the next sixteenth with uniform acceleration until the velocity was again doubled, and so on. Oresme found that in this case the total distance (or qualitas) is to that of the first half of the time as seven is to two.110 This is equivalent to summing geometrically to infinity the series i + | + j + A + A + A + A + "» See Wieleitner, "Ueber den Funktionsbegriff," pp. 231-35. «"Wieleitner, Ibid., p. 235.

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Calculator and Oresme were not the only Scholastics who were concerned with such infinite series. In the anonymous tract written before 1390, A est unum calidum,xn some of their results on series were rediscovered. More than two centuries later, in 1509, similar work appeared in the Liber de triplici motu of Alvarus Thomas, at Paris. This book was intended to serve as an introduction to the Liber calculationum of Suiseth and in it are handled several of the series given earlier by Calculator and Oresme. However, the author went on to similar but more complicated cases, in which he found (in the manner of Calculator and Oresme) the sums to infinity of the series 1 MI I H 1-1.12 l i oHIO-i 1 I Z i l i ? ! 1 4'2 8 ' 2» 16 ' 2' • • •ana 1 3 ' 2 6 ' 2' + 12 ' 2' -f- . . . , these sums being \ and ^r respectively.112 Alvarus Thomas remarked that innumerable other such series can be found. In some of these the sums involve logarithms, so that he could not arrive at the sum exactly, but gave it approximately between certain integers. Thus he said that the sum o f l + - j - | + | - ^ + ^ • ^ -f • • • (which results in 2 + log 2) lies between 2 and 4.113 It must be remembered that these infinite series were not handled by the Scholastic philosophers as they are now in the calculus, for they were given rhetorically, rather than by means of symbols, and were bound up with the concept of the latitude of forms. Furthermore, the results were found either through verbal arguments or geometrically from the graphical representation of the form, rather than by means of arithmetic considerations based upon the limit concept. Had their work been more closely associated with the geometrical procedures of Archimedes and less bound up with the philosophy of Aristotle, it might have been more fruitful. It could then have anticipated Stevin and the geometers of the seventeenth century in the elimination, from the classical method of exhaustion, of the argument by a reductio ad absurdum, and in the substitution of an analysis suggestive of the limit idea. The fourteenth-century work at Paris and Oxford on the latitude 111 Wieleitner, "Zur Geschichte der unendlichen Reihen," p. 167; cf. also Duhem, Études sur Léonard de Vinci, III, 474—77. 111 Wieleitner, "Zur Geschichte der unendlichen Reihen"; Duhem, Éludes sur Léonard de Vinci, IH, 531 ff. 111 Wieleitner, "Zur Geschichte der unendlichen Reihen," pp. 161-64; see also Duhem, Études sur Léonard de Vinci, m , 540-41.

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of forms and related topics was not forgotten during the subsequent decline of Scholasticism, for the doctrines spread throughout the Italian universities.114 In particular, they were taught at Pavia, Bologna, and Padua by Blasius of Parma.116 He, in turn, was referred to a century later by Luca Pacioli and Leonardo da Vinci ; and Nicholas of Cusa was very likely influenced by him.11' The work of Calculator and Oresme was much admired by the men of the fifteenth and sixteenth centuries, several printed editions of both the Liber calculationum and the De latitudinibus formarum, as well as commentaries on each, appearing in the interval from 1477 to 1520.117 In the De subtilitate of Cardan, Calculator is classed among the great, along with Archimedes, Aristotle, and Euclid.118 Galileo, we have said, gave in his dynamics a geometrical demonstration almost identical with that of Oresme, and in another place referred to Calculator and Hentisbery on activity in resisting media.11' Descartes likewise, in his attempt to determine the laws of falling bodies, employed arguments closely resembling those of Oresme.120 As late as the end of the seventeenth century the reputation of Calculator was such that Leibniz on several occasions referred to him as almost the first to apply mathematics to physics and as one who introduced mathematics into philosophy.121 In spite of the reputation maintained by the great proponents of the doctrine of the latitude of forms, however, the type of work they represented was not destined to be the basis of the decisive influence in the development of the methods of the calculus. The guiding Duhem, Études sur Leonard de Vinci, m , 481-510. See Thomdike, History of Magic, IV, 65-79. See Amodeo, "Appunti su Biagio Pelicani da Parma," III, 540-53. u ' The Liber calculalionum was published at Padua in 1477 and 1498, and again at Venice in 1520; De latitudinibus formarum appeared at Padua in 1482 and 1486, at Venice in 1505, and at Vienne in 1515. "» Cardan, Opera, i n , 607-8. m "Secunda dubitatio: quomodo se habent primae qualitates in activitate et resistentia. De hac re lege Calculatorem in tractatu De reactione, Hentisberum in sophismate." See Opere, I, 172. Œuvres, X , 58-61. m "Quis refert, primum prope corum, qui mathesin ad physicam applicarunt, fuisse Johan. [sic] Suisset calculatorem ideo, scholasticis appellatum": Letter to Theophilus Spizelius, April 7/17, 1670. Opera omnia (Dutens), V, 347. "Vellem etiam edi scripta Suisseti, vulgo dicti calculatoris, qui Mathesin in Philosophicam Scholasticam introduit, sed ejus scripta in Cottonianis non reperio." Letter to Thomas Smith, 1696. Opera omnia (Dutens), V, 567. See also Opera Omnia (Dutens), V, 421. U4 m

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principles were to be supplied by the geometry of Archimedes, although these were to be modified by kinematic notions derived from the quasi-Peripatetic disputations of the Scholastic philosophers on the subject of variation. As early as the beginning of the fifteenth century, the mensurational science and mathematics of Archimedes were becoming more influential, so that men like Blasius of Parma in some instances preferred his explanations to those of Aristotle. Blasius of Parma is said to have written on infinitesimals122 as well as upon other mathematical topics, but no such works are known to be extant. We do, however, have the work of Cardinal Nicholas of Cusa, which illustrates the influence both of the Scholastic speculations (perhaps through Blasius) and the work of Archimedes, as well as another tendency which grew up, particularly on German soil, in the fifteenth century. With the decline of the extremely rational and rigorous thought of Scholasticism (the overpreciseness of which was anathema to the growing Humanism of the time),121 there was a trend toward Platonic and Pythagorean mysticism.124 This may well have been in some measure responsible for the prevalence of mysticism12" at the time. On the other hand, the development of science in Italy, culminating in the dynamics of Galileo, owed much to the mathematical philosophy of Plato and Pythagoras.12' From the point of view of the rise of the calculus, this pervasive Platonism exerted for several centuries a not altogether unfortunate influence upon mathematics, for it allowed to geometry what Aristotle's philosophy and Greek mathematical rigor had denied—the free use of the concepts of infinity and the infinitesimal which Platonic and Scholastic philosophers had fostered. Nicholas of Cusa was not a trained mathematician, but he was certainly well acquainted with Euclid's Elements and had read Archimedes' works. Although he was primarily a theologian, mathematics nevertheless constituted for him, as for Plato, m Hoppe ("Geschichte der Infinitesimalrechnung," p. 179) says " E r ist also in dieser Beziehung ein Vorläufer von Cavalieri, von dem ihn 148 Jahre trennen." Blasius died in 1416, but probably Hoppe has in mind the 1486 edition of Blasius' Questiones, which is separated from Cavalieri's Geometría indivisibilibus of 1635 by about 148 years. "» Thomdike, History of Magic, HI, 370. m Lasswitz, Geschichte der Atomistik, I, 264-65; see also Burtt, The Metaphysical Foundations of Modern Physical Science, pp. 18-42. 114 Cf. Thomdike, Science and Thought in the Fifteenth Century, p. 22. a ' Strong, Procedures and Metaphysics, pp. 3-4; see also Cohn, Geschichte des Unendlichkeitsproblems, p. 95.

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the basis of his whole system of philosophy.14' For him mathematics was not restricted, as for Aristotle, to the science of quantity, but was the necessary form for an interpretation of the universe.128 Aristotle had insisted on the operational character of mathematics and had rejected the metaphysical significance of number, but Cusa revived the Platonic arithmology.12' He again associated the entities of mathematics with ontological reality and restored to mathematics the cosmological status which Pythagoras had bestowed upon it. Furthermore, the validity of its propositions was regarded by him as established by the intellect, with the result that the subject was not bound by the results of empirical investigation. This view of mathematics as prior to, or at least independent of, the evidence of the senses encouraged speculation and allowed the indivisible and the infinite to enter, so long as no inconsistency in thought resulted. Such an attitude enriched the subject and eventuated in the methods of the calculus, but this advantage was gained at the expense of the rigor which characterized ancient geometry. The logical perfection in Euclid found no counterpart in modern mathematics until the nineteenth century. From the point of view of the development of the calculus, this was not an unfortunate situation, for the infinite and the infinitely small were allowed almost free reign, even though it was about four hundred years before a logically satisfactory basis for these notions was found and the limit concept was definitely and rigorously established in mathematics. In spite of Aristotle, the infinitely large and the infinitely small had been employed somewhat surreptitiously by Archimedes in his geometry, and the Scholastics had freely discussed these notions in their dialectical philosophy; but with Nicholas of Cusa these ideas became, together with an unnecessarily large admixture of Pythagorean and theological mysticism,130 a recognized part of the subject matter of mathematics. The definitions given by Nicholas of Cusa for the infinitely large (that which cannot be made greater), and for the infinitely small m

Schanz, Der Cardinal Nicolaus von Cusa als Mathemaliker, pp. 2, 7. Lob, Die Bedeutung der Mathematik fiir die Erkenntnislekre des Nikolaus von Kues, pp. 36-37; cf. also Max Simon, Cusanus als Mathemaliker. "» Cf. L. R. Heath, The Concept of Time, p. 83. 1,0 Cohn, Gesckichte des Unendlichkeitsproblems, p. 127. 118

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(that which cannot be made smaller)1*1 are unsatisfactory, and his mathematical demonstrations are not always unimpeachable. His work is significant, however, in that it made use of the infinite and the infinitesimal, not merely as potentialities, as had Aristotle, but as actualities which are the upper and lower bounds of operations upon finite magnitudes. Just as the triangle and the circle were to Nicholas of Cusa the polygons with the smallest and the greatest number of sides, so also zero and infinity were to him the lower and upper bounds of the series of natural numbers.1*2 His view was colored, moreover, by his favorite philosophical doctrine—derived, perhaps, from the Scholastic quasi-theological disputations on the infinite—that a finite intelligence can approach truth only asymptotically. Consequently, the infinite was the source and means, and at the same time the unattainable goal, of all knowledge.115 Cusa was led by this attitude to view the infinite as a terminus ad quern to be approached only by going through the finite"4—an idea which represents a striving toward the limit concept185 which was to be developed during the next five hundred years. Upon the basis of this view, Nicholas of Cusa proposed a characteristic quadrature of the circle. If, contrary to Greek ideas of congruence, the circle belongs among the polygons as the one with an infinite number of sides and with apothem equal to the radius, its area may be found by the same means as that employed for any other polygon: by dividing it up into a number (in this case an infinite number) of triangles,136 and computing the area as half the product of the apothem and the perimeter. Nicholas of Cusa added to this explanation of the measurement of the circle an Archimedean proof, using inscribed and circumscribed polygons, and a proof by the reductio ad absurdum; but when his method was used later by Stevin, Kepler, and others, the Archimedean proof was abandoned, an elementary equivalent of the limit concept being considered sufficient. As Plato had opposed the Democritean atomic doctrine and yet attempted to link the continuum with indivisibles, so also Cusa objected to Epicurean atomism, but at the same time regarded the line 1,1 m Lorenz, Das Unendliche bet Nicolaus von Cues, p. 36. Lorenz, op. cit., p. 2. 134 Lasswitz, Geschichte der Alomistik, I, 282. ™ Colin, op. cit., p. 87. 114

Cf. Simon, "Zur Geschichte und Philosophic der Diiferentialrechnung," p. 116. Schanz, Nicolaus von Cusa, pp. 14-15.

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as the unfolding of a point. In a similar manner, also, he held that while continuous motion is thinkable, in actuality it is impossible,1'7 inasmuch as motion is to be regarded as composed of serially ordered states of rest.158 These views call to mind the modern mathematical continuum and the so-called static theory of the variable, but to Nicholas of Cusa must not be ascribed any of the precision of statement which now characterizes these matters. He did not make clear in what manner the transition from the continuous to the discrete is made. He asserted that although in thought the division of continuous magnitudes, such as space and time, may be continued indefinitely, nevertheless in actuality this process of subdivision is limited by the smallness of the parts obtained—that is, of the atoms and instants.159 The looseness of expression found toward the middle of the fifteenth century in the formulation by Nicholas of Cusa of his views as to the nature of the infinitesimal and the infinitely large was paralleled about two centuries later by a similar lack of clarity in the multifarious methods of indivisibles to which they to some extent gave rise. These latter procedures, in turn, led to the differential and the integral calculus. It is not to be concluded, however, that the views of Cusa are to be construed as indicating any renaissance in mathematics, nor as heralding the rise of the new analysis. The cardinal, absorbed as he was in matters concerning church and state, did not contribute any work of lasting importance in mathematics. Regiomontanus was perfectly justified in his criticism140 of the repeated attempts which were made by Cusa to square the circle. Nevertheless, as Plato's thought may have been instrumental in the use of infinitesimals made by Archimedes in his investigations preliminary to the application of the rigorous method of exhaustion, so also the speculations of Nicholas of Cusa may well have induced mathematicians of a later age to employ the notion of the infinite in conjunction with the Archimedean demonstrations. Leonardo da Vinci, who was strongly influenced, through Nicholas L. R. Heath, The Concept of Time, p. 82. Cohn, Gesckichie des Unendlichkeits problems, p. 94. L. R. Heath, The Concept of Time, p. 82; Cohn, op. cil., p. 94; Lasswitz, Geschichte der Atomistik, II, 276 ff. 110 See Kaestner, Geschichte der Mathematik, I, 572-76.

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of Cusa, by Scholastic thought,141 and who was acquainted also with the work of Archimedes, is said to have employed infinitesimal considerations in finding the center of gravity of a tetrahedron by thinking of it as made up of an infinite number of planes. We cannot, however, be sure of his point of view.142 Cusa's ideas were more clearly expressed in the work of Michael Stifel, who held that a circle may correctly be described as a polygon with an infinity of sides, and that before the mathematical circle there are all the polygons with a finite number of sides, just as preceding an infinite number there are all the given numbers.143 Somewhat later François Viète also spoke of a circle as a polygon with an infinite number of sides,144 thus showing that such conceptions were widely held in the sixteenth century. The fullest expression of Nicholas of Cusa's mathematical thoughts on the infinite and the infinitesimal, however, are found in the work of Johann Kepler, who was strongly influenced by the cardinal's ideas—speaking of him as divinus mihi Cusanus—and who was likewise deeply imbued with Platonic and Pythagorean mysticism. It was probably the imaginative use by Cusa of the concept of infinity which led Kepler to his principle of continuity—which included normal and limiting forms of a figure under one definition, and in accordance with which the conic sections were regarded as constituting a single family of curves.146 Such conceptions often led to paradoxical results, inasmuch as the notion of infinity had not yet received a sound mathematical basis; but Kepler's bold views suggested new paths which were be to very fruitful later. Kepler lived a century and a half after the time of Cusa. Moreover, unlike his predecessor, although his early training was in theology, he was primarily a mathematician. He was thus able, to the fullest extent, to take advantage Duhem, Études sur Léonard de Vinci, H, 99 ff. Libri, Histoire des sciences mathématiques en Italie, III, 41 ; Duhem, Études sur Léonard de Vinci, 1,35-36. The former asserts that Leonardo used indivisibles; the latter questions this. Duhem is wrong when he says in this connection (cf. Les Origines de la statique, H, 74) that Archimedes had restricted himself to plane figures in his work on the center of gravity, since in the Method the centers of gravity of segments of a sphere and of a paraboloid are determined. 143 Cf. Gerhardt, Geschichte der Mathematik in Deutschland, pp. 60-74. 1« Moritz Cantor, Vorlesungen, II, 539-40. 145 This is indicated by his statement, ". . . dico, lineam rectam esse hyperbolarum obtussissimam. Et Cusanus infinitum circulum dixit esse lineam rectam. . . . Plurima talia sunt, quae analogia sic vult efferi, non aliter." Kepler, Opera omnia, II, 595. 10

1U

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of the accessibility of Archimedes' works resulting from their translation and publication during the middle of the sixteenth century. It has been remarked that the medieval period added little to the classical Greek works in geometry or to the theory of algebra. Its contributions were chiefly in the form of speculations, largely from the philosophical point of view, on the infinite, the infinitesimal, and continuity, as well as of new points of view with reference to the study of motion and variability. Such disquisitions were to play a not insignificant part in the development of the methods and concepts of the calculus, for they were to lead the early founders of the subject to associate with the static geometry of the Greeks the graphical representation of variables and the idea of functionality. Both Newton and Leibniz, as well as many of their predecessors, sought for the basis of the calculus in the generation of magnitudes— a point of view which may be regarded as the most notable contribution of Scholastic philosophy to the development of the subject. However, the sound mathematical basis for the seventeenth-century elaboration of the infinitesimal procedures into a new analysis was supplied by the mensurational treatises of Archimedes which had been composed much earlier. These had become scattered and some had been lost, although a number of them were familiar to the Arabs, and although Archimedean manuscripts were known in the Scholastic period. 146 No significant additions to his results were made until after the appearance of printed editions of his treatises. In 1543 the Italian mathematician, Nicolo Tartaglia, published at Venice portions of Archimedes' work, including that on centers of gravity, the quadrature of the parabola, the measurement of the circle, and the first book on floating bodies. The editio princeps of Venatorius appeared at Basel in 1544, and in 1558 the important translation by Federigo Commandino appeared at Venice. Commandino himself composed a treatise in the Archimedean manner—Liber de centro gravitatis solidorum—showing that the study of the methods of the great Syracusan mathematician had advanced to the point where new contributions could be made. Shortly after this, however, numerous a t t e m p t s were made by Kepler and others to replace the almost tediously rigorous See Thorndike and Kibre, Catalogue of Incipits, for citations of some dozen manuscripts of the twelfth, thirteenth, fourteenth, and fifteenth centuries.

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arguments of Archimedes by new methods which should be equivalent to the older ones but at the same time possess a simplicity and an ease of application to new problems which was lacking in the method of exhaustion. It is precisely the search for such substitutes which was to lead within the next century, with the help of the Scholastic speculations, to the methods of the calculus. For the sake of convenience and unity, we shall make these anticipations the subject of the following chapter.

IV. A Century of Anticipation

T

HE DEVELOPMENT of the concepts of the calculus may be considered to have begun with the Pythagorean effort to compare—through the superposition of geometrical magnitudes—lengths, areas, and volumes, in the hope of thus associating with each configuration a number. Thwarted in this by the problem of the incommensurability of such magnitudes, it was left for later Greek geometers to circumvent such direct comparison through the method of exhaustion of Eudoxus. Through this method the need for the infinite and the infinitesimal had been obviated, although such notions had been considered by Archimedes as suggestive heuristic devices, to be used in the investigation of problems concerning areas and volumes which were preliminary to the intuitively clear and logically rigorous proofs given in the classical geometrical method of exhaustion. During the later Middle Ages the conceptions of the infinite and the infinitesimal, and the related ideas of variation and the continuum, were more freely discussed and utilized. Because, however, the interest in such speculations remained less closely associated with geometry than with metaphysics or science, or even with attempts to explain the possibility of natural magic,1 the primary influence leading to the calculus was not supplied by the doctrines of the Scholastic philosophers but by the greater enthusiasm for the methods of Archimedes which was manifest in the late sixteenth century and which continued throughout the whole of the century following. Whereas a multiplicity of printed editions of the works of Jordanus Nemorarius, Bradwardine, Calculator, Oresme, Hentisbery, and other medieval scholars had appeared in the late fifteenth and early sixteenth centuries, there arose toward the middle of the latter century a strong opposition, illustrated by the attitude of Ramus, to Aristotelianism and Scholastic methodology.2 It was during the height of this reaction that the works of Archimedes appeared in numerous editions and, admiring Archimedes, 1

Thomdike, History of Magic, HI, 371. * Johnson and Larkey, "Robert Recorde's Mathematical Teaching and the AntiAristotelian Movement."

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the men of the time refused to recognize the work of the Middle Ages.* In the mathematics of this period, nevertheless, the intrusion of influences from the Scholastic age is easily discernible in the attempts to reconcile with the thought of Archimedes the newly discovered infinitesimal methods which were fostered. There is noticeable in this development, as well, a tendency foreign alike to Archimedes and to the Scholastic period—a deeper interest in the Arabic algebra which had been developing in Italy, in which the concept of infinity did not figure. The algebra of Luca Pacioli's Summa de arithmetica was not greatly advanced over that to be found in Leonardo of Pisa almost three hundred years earlier,4 but in the sixteenth century the subject was assiduously studied again. Before 1545 the cubic had been solved by Tartaglia and Cardan, and the quartic by Ferrari; and thereafter a freer use of irrational, negative, and imaginary numbers was made by Cardan, Bombelli, Stifel, and others. The Greeks had not regarded irrational ratios as numbers in the strict sense of the word, and the attitude in the medieval period had been similar. Bradwardine asserted that an irrational proportion is not to be represented by any number; 6 and Oresme, in discussing the popular question of whether the celestial motions were commensurable or incommensurable, concluded that geometry favored the latter but arithmetic the former. 6 The Hindus and the Arabs, on the other hand, had not clearly distinguished between rational and irrational numbers. On adopting the Hindu-Arabic algebra, the sixteenth-century mathematicians had continued to employ the irrational ratio. They now recognized this as a number, but they stigmatized it, following Leonardo of Pisa, as a numtrus surdus, and they continued to interpret it geometrically as a ratio of lines.7 Negative quantities, admitted by the Hindus but not by the Greeks or the Arabs, were accepted in the sixteenth century as numeri falsi or ficli, but in the following century these were recogDuhem, Les Origines de la statique, I, 212. 4 Cajori, History of Mathematics, p. 128 Hoppe, "Zur Geschichte der Infinitesimalrechnung," p. 158. • Thorndike, History of Magic, III, 406. 7 See Pringsheim, "Nombres irrationnels et notion de limite," pp. 137-40; Fink, A Brief History of Mathematics, pp. 100-1. However, Kepler, as late as 1615 spoke of the irrational as "ineffable." Opera omnia, IV, 565. 3

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nized as numbers in the strict sense of the word. 8 Imaginary numbers also were regularly employed after the sixteenth century, although they continued to occupy an anomalous position in mathematics until the time of Gauss. Such generalizations of number, although not at the time based upon satisfactory definitions, were influential later in leading to the limit concept and to the arithmetization of mathematics. More important than this, in the development of the algorithm of the calculus, was the systematic introduction, during the later sixteenth century, of symbols for the quantities involved in algebraic relations. As early as the thirteenth century, letters had been used as symbols for quantities by Jordanus Nemorarius in his science and mathematics. Their establishment as symbols of the abstract quantities entering into algebra, however, was largely the work of the great French mathematician François Viète, 9 who used consonants to represent known quantities and vowels for those unknown. He distinguished arithmetic, or logistica numerosa, from algebra, or logistica speciosa, thus making the latter a calculation with letters rather than with numbers alone. This literal symbolism was absolutely essential to the rapid progress of analytic geometry and the calculus in the following centuries, 10 for it permitted the concepts of variability and functionality to enter into algebraic thought. The improved notation led also to methods which were so much more facile in application than the cumbrous geometrical procedures of Archimedes, of which they were modifications, that these methods were eventually recognized as forming a new analysis —the calculus. The period during which this transformation took place may be considered as the century preceding the work of Newton and Leibniz. 11 Numerous translations of the works of Archimedes had been made during the middle of the sixteenth century, and soon after this mathe• See Fine, Number System, p. 113; Paul Tannery, Notions historiques, pp. 333-34; Fehr, " L e s Extensions de la notion de nombre dans leur développement logique et historique." " Cajori, History of Mathematics, p. 139. 10 Cf. Karpinski, " I s There Progress in Mathematical Discovery?" p. 47. u For a comprehensive account of the methods developed during this period, see Zeuthen, Geschichte der Mathematik im XVI. und XVII. Jahrhundert. Zeuthen's account is an outline of the development of the subject in this interval, rather than of fundamental concepts, and so includes a large amount of mathematical detail.

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maticians had reached the point at which original contributions could be made to the classical work of the Greeks. This is evidenced by the fact that Commandino in 1565 published a work of his own on centers of gravity. In this he proved, among other things, that the center of gravity of a segment of a paraboloid of revolution is situated on the axis two-thirds of the distance from the vertex to the base.12 This was a proposition which Archimedes had, incidentally, demonstrated by infinitesimals in a treatise, the Method, which was apparently not known at that time. Commandino's proof followed the orthodox style of the method of exhaustion. Such extensions, however, had perhaps less influence on the development of the calculus than did certain striking innovations in method introduced by succeeding mathematicians. Perhaps the first of such significant modifications is that advanced by Simon Stevin of Bruges in 1586, very nearly a century before the first printed work on the calculus by Leibniz, in 1684, and that of Newton in 1687. Stevin was essentially an engineer and a practical-minded scientist. For this reason he had perhaps less regard for the philosophy of science and the exigencies of mathematical rigor than for technological tradition and methodology.13 As a result, Stevin did not merely imitate, as had Commandino, Archimedes' use of the method of exhaustion: he accepted the direct portion of his characteristic proof as sufficient to establish the validity of any proposition that required it, without adding in every case the formal reductio ad absurdum required by Greek rigor. Furthermore, he frequently omitted, as we do in the integral calculus, one of the approximating figures which Archimedes had used, being satisfied with either the inscribed or the circumscribed figure only. Stevin demonstrated as follows (in his work on statics, in 1586) that the center of gravity of a triangle lies on its median. Inscribe in the triangle ABC a number of parallelograms of equal height, as illustrated (fig. 7). The center of gravity of the inscribed figure will lie on the median, by the principle that bilaterally symmetrical figures are in equilibrium (a principle used by Archimedes in proving the law of the lever, and also by Stevin in his well-known demonstration of the law of the inclined plane). However, we may inscribe in the triu u

Commandino, Liber de centra gravitatis solidorum, fol. 40°-41°. Strong, Procedures and Meiaphysics, pp. 91-113.

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angle an infinite number of such parallelograms, for all of which the center of gravity will lie on AD. Moreover, the greater the number of parallelograms thus inscribed, the smaller will be the difference between the inscribed figure and the triangle ABC. If, now, the "weights" of the triangles A BD and A CD are not equal, they will have a certain fixed difference. But there can be no such difference, inasmuch as each of these triangles can be made to differ by less than this from the sums of the parallelograms inscribed within them, which are equal. Therefore the "weights" of ABD and ACD are equal, and hence the center of gravity of the triangle ABC lies on the median AD.U

FIGURE 7 Exactly analogous demonstrations were given by Stevin of propositions on the centers of gravity of plane curvilinear figures, including the parabolic segment. These proofs given by Stevin indicate the direction in which the method of limits was developed as a positive concept. The Greek method of exhaustion had not boldly concluded, as did Stevin, that inasmuch as the difference could, by continued subdivision, be shown to be less than any given quantity, there could as a result be no difference. The Greeks felt constrained in every case to add the full reductio ad absurdum proof to show the equality. Stevin did not, of course, speak of the triangle as the limit of the sum of the Stevin, Hypomnemata mal chez Simon Stevin." 14

mathematica, IV, 57-58; cf. also Bosmans, "Le Calcul infinitési-

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inscribed parallelograms; but it would require only slight changes in his method—largely in the nature of further arithmetization and the use of greater precision in terminology—to recognize in it our modern method of limits. That Stevin regarded his approach to these problems as a significant modification of the method of Archimedes may be indicated in his proof of the proposition on the center of gravity of a conoidal segment —a proposition "the demonstration of which the ingenious and subtle mathematician Fredericus Commandinus gives in proposition 29 de solidorum centrobaricis, and which is arranged as follows in accordance M

A

r

F

B

Q

D

P

C

FIGURE 8 with our custom and method." 16 Circumscribe about the segment ABC two cylindrical segments FGBC and MLIK as illustrated (fig. 8). Now the centers of gravity of these cylindrical segments are, by the principle of the equilibrium of bilaterally symmetrical figures, at the midpoints, N and 0, of their axes, AH and HD, respectively; and the center of gravity of the entire circumscribed figure is at R such that NR = 2RO. Letting E be the point such that AE = 2ED, it can be shown that ER = ruAD. If, now, one similarly circumscribes about 16

Stevin, Hyfomnemaia mathemaiica, II, 75-76.

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the segment ABC four such cylindrical segments of equal height, the center of gravity of this circumscribed figure is found to lie above E at a point L such that EL = TIA D. Successively doubling the number of these cylinders, the center of gravity of the circumscribed figure remains always above E and will differ from E by TSAD, t s A D , and so on. Thus the center of gravity descends, approaching E more and more closely. Modem mathematics would now conclude that E is the limit of the center of gravity of the circumscribed figure and therefore is the center of gravity of the conoidal segment. Stevin, however, reached this conclusion only after cautiously observing t h a t similarly the center of gravity of the analogous inscribed figure ascended toward E in the same manner. The demonstration given by Stevin, in the above proposition in terms of the sequence A , t t , TW, f j , • • -, is comparable to that employed by Archimedes in his quadrature of parabola, in which the series l + i + rff + A + . . . figured. Both the Greek mathematician and the Flemish engineer, in their use of such sequences and series, stopped short of the limit concept. Neither thought of such a sequence or series as carried out to an infinite number of terms in the modern sense. Archimedes explicitly stopped with the nth term ^ ^ and added the term, | • representing the remainder of the series; Stevin used the word infinite in the Peripatetic sense of potentiality only—the sequence could be continued as far as desired, and the error consequently made as small as one pleased. For this reason Archimedes had been constrained to complete his work through the demonstration by a reductio ad absurdum; Stevin, although he adduced no such formal argument, had recourse also to supplementary demonstrations, such as the inclusion, in the above proposition on the conoid, of a second sequence approaching the point from the other side. This hesitation on the part of Stevin to accept as sufficient the notion of the limit of an infinite sequence is apparent also in a proposition in his hydrostatics, which represented perhaps his nearest approach to the method of limits. Here he supplemented the "mathematical demonstration" of propositions, carried out as above, by a "demonstration by numbers," suggested, perhaps, by the then recent Italian work in algebra and mensuration, and encouraged by the neglect of geometry

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in favor of arithmetic found in the Netherlands during the sixteenth century.16 In supplementing the proof 17 that the average pressure on a vertical square wall of a vessel full of water corresponds to the pressure at its mid-point, he gave an "example with demonstration." He subdivided the wall into 4 horizontal strips and noted that the force on each is greater than 0, tb-, TS, and TS units, and less than TV, TE, TS, and TZ units respectively; so that the total force is greater than A and less than -nr- If the wall is subdivided into 10 horizontal strips, the force is found similarly to be greater than -nns and smaller than -nnr units; on using 1,000 strips, it is determined as more than ^'o^.ooo u n ' t s > a n d less than i5Q°0Q05Q°0°0 units. 18 By increasing the number of strips, he then remarked, one may approach as closely as desired to the ratio one-half, thus proving that the force corresponds to that which would be observed if the wall were placed horizontally at a depth of half a unit.1® This "demonstration by numbers" would correspond exactly to that given in the calculus if Stevin had limited himself to one of his two sequences and had thought of the results given by his successive subdivisions of the wall as forming literally an infinite sequence with the limit 5. Stevin, however, shared not only the Greek view with regard to infinity but also, although to a lesser extent than did most of his mathematical contemporaries, the classical apotheosis of geometry, so characteristic of ancient mathematics, which was to prevail in the seventeenth century. Even he considered his arithmetic proof, outlined above, as merely a mechanical illustration to be distinguished from the general mathematical demonstration. 20 Perhaps the one tendency that did more than any other to conceal from mathematicians for almost two centuries the logical basis of the " Cf. Struik, "Mathematics in the Netherlands during the First Half of the XVIth Century." 17 Stevin, Ilypomnemaia mathematica, II, 121 ff. 13 It will be recalled that Stevin had introduced his use of decimal fractions, in De Thiende, which had appeared in 1585, the year before the publication of his work in statics. See Sarton, "The First Explanation of Decimal Fractions and Measures (1585)." 18 Stevin, Ilypomnemaia mathematica, II, 125-26; cf. also Bosmans, "Le Calcul chez Stevin," pp. 108-9. 20 "Mathematicae & mechanicae demonstrations a doctis annotatur differentia, neque injuria. Nam ilia omnibus generalis est, & rationem cur ita sit penitus demonstrat, haec vero in subjecto duntaxat paradigmate numeris declarat." Stevin, Hypomnemata mathematica, II, 154.

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calculus was the result of the attempt to make geometrical, rather than arithmetic, conceptions fundamental. This will be more true of Stevin's successors than it was of him. It must be borne clearly in mind, however, that although the logical basis of the calculus is arithmetic, the new analysis resulted largely from suggestions drawn from geometry. The procedures substituted by Stevin for the method of exhaustion constituted a marked step toward the limit concept. The extent of his influence on contemporary thought, however, is difficult to determine. The work in statics which contains his anticipations of the calculus appeared in Flemish in 1586. It was included also in the Flemish, French, and Latin editions of his mathematical works published in 1605-8, and in a later French translation of his works by Girard in 1634.21 With the exception of the last-mentioned edition, however, most of these were not easily accessible to mathematicians, 22 and, by the time of the French translation of 1634, there had already appeared in Italy and Germany a number of alternative methods which were destined to become much more widely known than those of Stevin. Nevertheless, the influence of the Flemish scientist is evident in the thought of a number of later mathematicians of the Low Countries. 23 Before we turn to these men, however, it may be well to indicate briefly the nature of the modifications of Archimedes' work which appeared in Italy and Germany shortly after Stevin had published his methods. Luca Valerio likewise attempted, in his De centro gravitatis solidorum of 1604, a methodization of Archimedes' procedure which should obviate the need for the reductio ad absurdum and yet retain the necessary rigor of demonstration. His change was not so sweeping as that of Stevin, and his view is not so closely related to the modern. He merely tried to substitute for the method of Archimedes a few general theorems which could be cited instead of carrying through the details of proof in each and every case. No attempt had been made in Greek geometry to establish such propositions which might simply be quoted in particular cases in lieu of the double reductio ad absurdum demonstration. 24 n

See the article by Sarton, "Simon Stevin of Bruges (1548-1620)"; and that by Bosmans, "Simon Stevin," for biographies of Stevin, analyses of his work, and extensive bibliographical references. n Bosmans, "Simon Stevin," p. 889. B Bosmans, "Sur quelques exemples de la méthode des limites chez Simon Stevin." " The Works of Archimedes, Introduction, p. cxliii.

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Valerio professed that it was the appearance of Commandino's work on centers of gravity which had encouraged him,26 as it had Stevin,** to attempt such a modification of the method of Archimedes. The significant generalization is found in the proposition that, given any figure in which the distance between points on opposite sides of a diameter vanishes, parallelograms can be inscribed and circumscribed in such a way that the excess of the circumscribed figure over that inscribed is less than any given area.27 This is proved by observing that the excess is in each case equal to the area of the parallelogram BF (fig. 9) and that for suitably chosen approximating figures this "is less than a given area." 28 Then Valerio assumed without proof that if the difference be-

tween the inscribed and the circumscribed figures is smaller than any given area, this will be true also of the difference between the curve and either of these figures. This geometrical reasoning is strikingly similar to that presented in many present-day elementary textbooks on the calculus. Valerio did not, however, regard the area of the curve as necessarily defined by the limit of the area of either the inscribed or the circumscribed figure, as the number of such parallelograms becomes infinite. This is a sophisticated arithmetical conception which ** Valerio, De centro gravitatis solidorum libri très, p. 1. 2 ® It is doubtful whether Valerio knew of the work of Stevin. See Bosnians, "Les Démonstrations par l'analyse infinitésimale chez Luc Valerio," p. 211. 27 Valerio, De centro gravitatis, p. 13. 28 "Sed parallelogrammum BF est minus superficie proposita." Ibid., p. 14.

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was not established until two centuries later. Nevertheless, Valerio was in a sense anticipating the limit concept, in geometrical form, to the extent of indicating the necessary condition for the existence of such a limit—viz., that the difference in these areas can be made less than any specified area. Having generalized, by the above demonstration, the method of inscribing areas, Valerio then proceeded to state general propositions which should replace, at least in so far as ratios were concerned, the argument by a reductio ad absurdum as found in the method of exhaustion. The intention of these may be stated in the following form: If four magnitudes, A, B, C, D, are given, and if two others, G and H, can be found which are at the same time greater or smaller than A and Q B by a magnitude less than any given magnitude, and if the ratio — is H A C A at the same time 6greater or smaller than the ratios — and —, then — B D B C = —.29 This proposition is significant not only as a methodization of Archimedes' procedure, but also as a vague striving toward the idea which is now expressed concisely by saying that the limit of a ratio of two variables is equal to the ratio of the limits of these variables. This latter concept is dependent, however, on considerations of the infinite —on speculations in which Valerio did not indulge, but which interested many of his contemporaries and successors, particularly those who combined mathematical investigations with theological interests. Johann Kepler had been educated with the intention of entering the Lutheran ministry, but had been forced to turn later to the teaching of mathematics as a means of earning his livelihood. This may account in part for the fact that, although he owed as much to Archimedes as had Stevin, the nature of his work is so markedly different from that of the engineer of Bruges. There is in the thought of Kepler a deep strain of mysticism which was lacking in the attitude of Stevin, of whom Kepler may have known.30 However, this speculative tendency on the part of M De ceniro gravitaUs, p. 69; see also Wallner, "Grenzbegriffes," p. 251.

M Bosnians ("Les Démonstrations sur l'analyse infinitésimale chez Luc Valerio," p. 211) holds that Kepler was evidently familiar with Stevin's work; and Hoppe ("Zur Geschichte der Infinitesimalrechnung," p. 160) that he may have known of it; Wieleitner ( " D a s Fortleben") believes that such an assumption is not justified.

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Kepler is perhaps with more justice to be ascribed to the PlatonicPythagorean influence, which had been strong in Europe in the previous century, 31 and which was so evident in the thought of Cusa, to whom Kepler owed at least part of his inspiration. 32 The belief that the universe was an ordered mathematical harmony, so strongly shown in Kepler's Mysterium cosmographicum, was combined with Platonic and Scholastic speculations on the nature of the infinite, giving him a modification of Archimedes' mensurational work which was to be a powerful influence in shaping the development of the calculus. The ancient Greek philosophers, in their search for unity in this universe of perplexing multiplicity, had failed for two reasons to bridge the gap between the curvilinear and the rectilinear: first, they banned the infinite from geometry; and second, they hesitated, following the discovery of the irrational, to pursue further the Pythagorean association of numerical considerations with geometrical configurations. The pious enthusiasm of Kepler, however, saw in this impasse but one more evidence of the handiwork of the Creator, who had established all things in harmony. God wished quantity to exist so that the comparison between a curve and a straight line might be made. This fact was made clear to him by the "divine Cusanus" and by others who had regarded the forms of the curve and the straight line as complementary, daring to compare the curve to God and the line to his creatures. "For this reason those who have tried to relate the Creator to his handiwork and God to man and divine judgments to human represent an occupation which is by no means more useful than that of those who seek to compare the circle with the square." 33 Under such inspiration, guided as well by the speculations of Cusa and Giordano Bruno on the infinite in cosmology,34 and fortified by the knowledge "that nature teaches geometry by instinct alone, even without ratiocination," 36 Kepler was led to develop his modification of the procedures of Archimedes. The task of writing a complete treatise on volumetric determinations seems to have been suggested to Kepler by the prosaic problem of determining the best proportions for a wine cask. The result was 31 Burtt, Metaphysical Foundations, pp. 44-52; Strong, Procedures and Metaphysics, pp. 164 ff. 32 Cf. Kepler, Opera omnia, I, 122; II, 490, 509, 595. 33 Ibid., I, 122. » I b i d . , II, 509. « I b i d . , IV, 612.

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the Nova stereometria, which appeared in 1615.36 This contains three parts, of which the first is on Archimedean stereometry, together with a supplement containing some ninety-two solids not treated by Archimedes. The second part is on the measurement of Austrian wine barrels, and the third on applications of the whole. Kepler opened his work on curvilinear mensuration with the simple problem of determining the area of the circle. In this he abandoned the classical Archimedean procedures. He did not substitute for these the limiting consideration proposed by Stevin and Valerio, but had recourse instead to the less rigorous but more suggestive approach of Nicholas of Cusa. Like Stifel and Viète, he regarded the circle as a regular polygon with an infinite number of sides, and its area he therefore looked upon as made up of infinitesimal triangles of which the sides of the polygon were the bases and the center of the circle the vertex. The totality of these was then given by half the product of the perimeter and the apothem (or radius) Kepler did not limit himself to the simple proposition above, but with skill and imagination applied this same method to a wide variety of problems. By looking upon the sphere as composed of an infinite number of infinitesimal cones whose vertices were the center of the sphere and whose bases made up the surface, he showed that the volume is one-third the product of the radius and the surface area. 38 The cone and cylinder he regarded variously: as made up of an infinite number of infinitely thin circular laminae (as had Democritus two thousand years earlier), as composed of infinitesimal wedge-shaped segments radiating from the axis, or as the sum of other types of vertical or oblique sections.39 The volumes of these he computed by the application of such views. In a similar manner he rotated a circle about a line and calculated, by infinitesimal methods, the volume of the anchor ring, or tore, thus generated. 40 This determination was equivalent to an application, for a special case, of the classical theorem of Pappus, later called Guldin's rule. Kepler then extended his work to solids not treated by the ancients. The areas of the segments cut from a circle by a chord he rotated about this chord, obtaining solids " Followed, about a year later, by a popular German edition. The Latin appears, with notes, in Volume IV, Opera omnia; the German in Volume V. n Opera omnia, IV, 557-58. » Ibid., IV, 564, 568, 576 ff. « Ibid., IV, 575-76, 582-83. » Ibid., IV, 563.

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which he designated characteristically as apple or citron-shaped, according as the generating segment was greater or less than a semicircle.41 The volumes of these and other solids he likewise calculated by his infinitesimal methods. To Willebrord Snell, the editor of Stevin's works, Kepler proposed, as a challenge, the determination of the solids obtained similarly by the rotation of segments of conic sections.42 This problem was significant in the later work of the Italian mathematician Cavalieri. Some of Kepler's summations are remarkable anticipations of results found later in the integral calculus.43 In his well-known Astronomía nova of 1609, for example, there is a computation44 resembling that which is expressed in modern notation as Jo sin 6 dd = 1 — cos 6. Other calculations in this work correspond to approximations to elliptic integrals,46 in one of which irla + b) is given as the approximate length of the ellipse with the semi-axes a and b.4t Kepler, however, was far from clear on the point of the basic conceptions involved, with the result that his work is not free from errors.47 He generally spoke of surfaces and volumes as made up of infinitesimal elements of the same dimension, but occasionally he lapsed into the language of indivisibles, which his successor Cavalieri was to find so congenial. In one place he spoke of the cone as though composed of circles,48 and in the work by which he arrived in his astronomy at his famous second law he regarded the sector of an ellipse as the sum of its radius vectors. 48 Kepler appears not to have distinguished clearly between proofs by means of the method of exhaustion, by ideas of limits, by infinitesimal elements, or by indivisibles. The conceptions which he held in his dem41 Wolf (History of Science, Technology, and Philosophy in the Sixteenth and Seventeenth Centuries, pp. 204-5) has correctly pointed out that the word ciiriutn, which Kepler used in this connection, is properly translated as "gourd"; but he has failed to add that Kepler himself, in his German edition, translated it as "citron." Opera omnia, V, 526. ö Ibid., IV, 601,656. 43 See Struik, "Kepler as a Mathematician," in Johann Kepler, 1571-1630. This volume also contains an excellent bibliography of Kepler's works, by F. E. Brasch. 44 See Opera omnia, III, 390; cf. also Gunther, "Über ein merkwürdige Beziehung zwischen Pappus und Kepler"; Eneström, "Über die angebliche Integration einer trigonometrischen Funktion bei Kepler." 45 Cf. Struik, "Kepler as a Mathematician," p. 48; Zeuthen, Geschichte der Mathematik im XVI. und XVII. Jahrhundert, pp. 254-55. 44 Opera omnia, III, 401. 47 Moritz Cantor, Vorlesungen, II, 753. 48 "Nam conus est hic veluti circulus corporatus." Opera omnia, IV, 568. 48 Opera omnia, III, 402-3.

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onstrations are a far cry from the notions held by the ancient geometers. The Greek thinkers saw no way of bridging the gap between the rectilinear and the curvilinear which would at the same time satisfy their strict demands of mathematical rigor and appeal to the clear evidence of sensory experience. Fortified by the scholastic disputations on the categorical infinite and by Platonic mathematical speculations, Kepler followed Nicholas of Cusa in resorting to a vague "bridge of continuity" which finds no essential difference between a polygon and a circle, between an ellipse and a parabola, between the finite and the infinite, between an infinitesimal area and a line.50 This striving for an expression of the idea of continuity constantly reappears throughout the period of some fifty years preceding the formulation of the methods of the calculus. Leibniz himself, like Kepler, frequently fell back upon his so-called law of continuity when called upon to justify the differential calculus; and Newton concealed his use of the notion of continuity under a concept which was empirically more satisfying, though equally undefined—that of instantaneous velocity, or fluxion. Kepler's Doliometria, or Stereometria doliorum, exerted such a strong influence in the infinitesimal considerations which followed its appearance, and which culminated a half century later in the work of Newton, that it has been called, with perhaps pardonable exaggeration, the source of the inspiration for all later cubatures. 61 Before passing on to these anticipations of the integral calculus, there should be cited a contribution which Kepler made to the thought leading to the differential calculus. The subject of the measurement of wine casks had led Kepler to the problem of determining the best proportions for these.62 This brought him to the consideration of a number of problems on maxima and minima. In the Doliometria he showed, among other things, that of all right parallelepipeds inscribed in a sphere and having square bases, the cube is the largest ;53 and that of all right circular cylinders having the same diagonal, that one is greatest which has the diameter and altitude in the ratio of to 1." These results were obtained by making up tables in which were 60

Cf. Taylor, "The Geometry of Kepler and Newton." Moritz Cantor, Vorlesungen, II, 750. Kepler discovered, incidentally, that the Austrian barrels approximated very closely the desired proportions. M Opera omnia, IV, 607-9. " Ibid., IV, 610-12. M

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listed the volumes for given sets of values of the dimensions, and from these selecting the best proportions. An inspection of such tables showed him an interesting fact. He remarked that as the maximum volume was approached, the change in the volume for a given change in the dimensions became smaller. Oresme, several centuries earlier, had made a similar observation, but had expressed it differently. Oresme had noticed that for a form which was represented graphically by a semicircle, the rate of change was least at the maximum point. This thought appeared again, in the seventeenth century, in the methods of the French mathematician Fermat. Whether Fermat was influenced in this direction by Kepler or Oresme is problematical; but a comparison of the distinctly different points of view which the latter men represent will aid later in understanding Fermat's approach. Kepler had made his remark upon the basis of numerical considerations. He was, moreover, more particularly concerned with static considerations as found in Greek geometry and in methods of indivisibles. He consequently expressed himself in terms of increments and decrements near the maximum point. On the other hand, the medieval problem of the latitude of forms and the graphical representation of continuous variability had led Oresme to state the conclusion in terms of the rate of change. The latter view has been made fundamental in mathematics through the concept of the derivative; but it is Kepler's mode of expression which appeared in the work of Fermat. Although the Scholastic views on variation played a significant role in the anticipations of the calculus, the static approach of Kepler predominated. Increments and decrements, rather than rates of change, were the fundamental elements in the work leading to that of Leibniz, and played a larger part in the calculus of Newton than is usually recognized. The differential became the primary notion and it was not effectively displaced as such until Cauchy, in the nineteenth century, made the derivative the basic concept. Twenty years after the publication of the Stereometria doliorum of Kepler there appeared in Italy a work which rivaled it in popularity. So famous did the Geometria indivisibilibus of Bonaventura Cavalieri become that it has been maintained, with some justice, that the new analysis took its rise from the appearance, in 1635, of this book.66 To " Leibniz, The Early Mathematical Manuscripts,

trans, by Child, p. 196.

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what extent this is indebted to the earlier work of Kepler is difficult to determine. Cavalieri emphatically denied any inspiration from Kepler's method, other than "the names of a few solids and the admiration which frequently sets philosophers to reflecting."66 It is not improbable, however, that the influence of Kepler upon Cavalieri may have resulted indirectly from the correspondence of both of these men with Galileo.67 A work upon indivisibles which Galileo planned to write never appeared, but his views upon the subject are clearly brought out in his classic treatise, the Two New Sciences, which was published three years after the Geometría of Cavalieri. Galileo's opinions resemble strongly those expressed by his pupil, Cavalieri, and may well have been the source of the latter's inspiration. 68 It may be well, therefore, to examine at this point the views of Galileo, the teacher. The forces molding the thought of Galileo and Cavalieri did not differ greatly from those which had shaped the ideas of Kepler. These men had mastered the Greek geometric methods, but they all betray the effects of Scholastic speculations and the Platonic view of mathematics which had exerted such a strong influence since the time of Nicholas of Cusa. Both Galileo and Cavalieri were probably acquainted with the modifications of the method of Archimedes by Valerio. Galileo referred to Valerio several times, in the Two New Sciences, as the great geometer and as the new Archimedes of his age.69 Galileo gave in this work also an Archimedean demonstration of the quadrature of the parabola 60 and included in an appendix some work on centers of gravity in the manner of Commandino and Valerio.61 Nevertheless both he and Cavalieri appear to be more significantly indebted, in attitude and method, to the later medieval speculations on motion, indivisibles, the infinite, and the continuum. 62 The influence of Scholastic thought is clearly evidenced in the case of Galileo by his early writings. In these he considered, among other things, the Peripatetic doctrines of matter and form and of causes and qualities, and the Scholastic questions of intension and remission and of action and reaction.63 On the latter he referred specifically to works " Exercilaliones geometricae sex, pp. 237-38. " P a u l Tannery, Notions historiques, p. 341; Moritz Cantor, Vorlesungen, II, 774-75. « Paul Tannery, ibid., p. 341. Le opere di Galileo Galilei, VIII, 76, 184, 313. •oOpere, VIII, 181 ff. " Opere, VIII, 313; I, 187-208. " Wallner, "Die Wandlungen des Indivisibiliensbegriffs von Cavalieri bis Wallis." "Opere, I, "Iuvenalia," 111 ff., 119 ff., 126 ff.

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by Calculator and Hentisbery.*4 In dynamics he made fundamental the doctrine of impetus, which had been suggested by the Scholastic philosopher Buridan in the fourteenth century and which was certainly familiar to Nicholas of Cusa, Leonardo da Vinci, and others of the fifteenth and sixteenth centuries.6® Whether Cusa's influence on Galileo was significant (as it was on Kepler and probably also on Giordano Bruno) is difficult to determine;66 but that the Scholastic discussions on motion were known to him will be very strongly indicated by an examination of one of the propositions in his Two New Sciences.

It will be recalled that Calculator and Hentisbery had demonstrated dialectically that the average velocity of a body moving with uniform acceleration is given by its velocity at the mid-point of the time interval. Oresme had given a geometrical demonstration of this proposition, in which he indicated that the area under the line representing the velocity was the measure of the distance. Galileo's demonstration of this proposition resembles strikingly that of Oresme. Let AB (fig. 10) represent the time in which the space CD is traversed by a body which M Ibid., I, 172. « Duhem, Études sur Léonard " Goldbeck, "Galileis Atomistik und ihre Quellen."

de Vinci,

Vol. Ili,

passim.

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starts from rest and is uniformly accelerated. Let the final speed be represented by EB. Then the lines drawn parallel to EB will represent the speeds of the body. It appears, then, that they may be interpreted also as the moments, or infinitesimal increments, in the distance covered by the moving body. Then the movements of the uniformly accelerated motion may be represented by the parallels of triangle AEB, whereas the parallels of the rectangle ABFG represent the corresponding moments of a body moving uniformly. But the sum of all the parallels contained in the quadrilateral A BFG is equal to the sum of those contained in the triangle AEB. Hence it is clear that the distances covered by the two bodies are equal, inasmuch as the triangle and the rectangle are equal in area if I is the midpoint of FG.67 Not only did Galileo reproduce with striking fidelity the diagram and argument of Oresme as outlined above, but he extended a remark made in this connection by Calculator and Ilentisbery. Galileo could have read in the works of these men, or in commentaries upon them, that in uniformly accelerated motion the space covered in the second half of the time is three times that covered in the first half.68 This observation Galileo extended to show that if we subdivide the time interval further, the distances covered, in each of these, will be in the ratio 1, 3, 5, 7, . . . 69 This is, of course, equivalent to the result expressed by the formula s = and is implied by the earlier work of the Scholastics. The difference between Galileo's demonstration and that of Oresme is largely one of completeness. Oresme had been satisfied to say merely that inasmuch as the triangles EI F and AIG are equal, the distances must be the same, thus implying the infinitesimal considerations necessary to demonstrate that the areas represent distances. The geometrical demonstrations of Oresme and Galileo are based upon the supposition that the area under a velocity-time curve represents the distance covered.70 Since neither one possessed the limit concept, each resorted, explicitly or implicitly, to infinitesimal considerations. Galileo expressed this view when he said that the moments, or small increments in the distance, were represented by the lines of the triangle and the "Opere, VIII, 208-9. 69 » Duhem, Études sur Léonard de Vinci, III, 480, 513. Opere, VIII, 210 ff. 70 Paul Tannery, "Notions historiques," pp. 338-39, mistakenly attributes this idea to Galileo, rather than Oresme.

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rectangle, and that these latter geometrical figures were in actuality made up of these lines. Galileo did not make clear how the transition from the lines as velocities to the same lines as moments is to be made. Oresme likewise had begged the question when he had represented instantaneous velocities by lines and yet had maintained that all velocities act through a time. Galileo and Oresme patently employed the uncritical mathematical atomism which has appeared among mathematicians of all ages—in Democritus, Plato, Nicholas of Cusa, Kepler, and many others. It has been suggested that Galileo was influenced in his use of indivisibles by his strong Pythagorean and Platonic approach to science,71 or by the revival in his day of interest in the atomism of Heron of Alexandria, in which Galileo failed to distinguish clearly between physical indivisibles and mathematical infinitesimals.72 He may equally well have been led to his views by the Scholastic discussions of infinitesimals. At any rate, in the dialogue of the first day, in the Two New Sciences, Galileo entered into an extended discussion of the subjects which had been so popular in the medieval period: the infinite, the infinitesimal, and the nature of the continuum. Galileo clearly admitted the possibility of the Scholastic categorematic infinity, but because of the numerous paradoxes to which this appeared to lead, Galileo concluded that "infinity and indivisibility are in their very nature incomprehensible to us." 73 Nevertheless, he made at least one trenchant observation upon this subject. It will be recalled that Calculator had remarked that there can be no ratio between an infinite magnitude and a finite one. Galileo asserted more generally "that the attributes 'larger,' 'smaller,' and 'equal' have no place either in comparing infinite quantities with each other or in comparing infinite with finite quantities." 74 In justifying this conclusion, Galileo indicated a significant shift of emphasis, for instead of considering the infinite from the point of view of magnitude, as had Aristotle and many medieval scholars, he focused attention, as had Plato, upon the infinite as multiplicity or aggregation. In this connection he indicated that the 71

Wiener, "The Tradition behind Galileo's Methodology"; cf. also Branschvicg, Les Étapes de la philosophic mathématique, p. 70. 72 See Schmidt, "Heron von Alexandria im 17. Jahrhundert"; also other articles on Heron in the same volume. 73 Opere, VIII, 76 ff. » Opere, VIH, 82 ff.

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infinite class of all positive integers could be put into a one-to-one correspondence with a part of this class—for example, with all the perfect squares. 7 ' This characteristic of infinite sets was rediscovered in the nineteenth century by Bolzano and later in that century was made fundamental in the establishment of the calculus upon a rigorously developed theory of infinite assemblages.76 In spite of the trenchant observations which Galileo made on the subject of the infinite, he felt strongly the inability of intuition to grasp this notion. He went so far as to conjecture that there might be a third possible type of aggregation between the finite and the infinite. This seems to have been suggested to him by the difficulties of making precise our vague ideas on the continuum. He maintained, contrary to Bradwardine, that continuous magnitudes are made up of indivisibles. However, inasmuch as the number of parts is infinite, the aggregation of these is not one resembling a very fine powder but rather a sort of merging of parts into unity, as in the case of fluids.77 This analogy is a beautiful illustration of the effort which men made to picture in some way the transition from the finite to the infinite. Galileo sought also to clarify to some extent the paradoxes on motion. This he did by regarding rest as an infinite slowness, thus resorting again to the vague feeling for the continuous to which Cusa and Kepler had sought to give expression. In enlarging upon this idea, Galileo applied to a falling body the argument which Zeno had given in the dichotomy and then answered this by an appeal to intuition in reversing the descent. In ascent, the body passes through an infinite number of grades of slowness, finally to come to rest. The beginning of motion is precisely the same, except that the order is reversed. 78 This argument constitutes a recognition—found also in Aristotle—of the similarity of the difficulties in the dichotomy and the Achilles. It represents, as well, an attempt to clarify the sense in which an infinite series may be said to have a sum. Later Newton in his calculus, when he spoke of his instantaneous velocities as prime or ultimate ratios, made an analogous appeal to the fact that the beginning and the end of motion are to be similarly conceived. Galileo appears not to have realized, as Newton did somewhat vaguely, that only in terms of the " Ibid., VIII, 78. 77

Kasner, "Galileo and the Modem Concept of Infinity," pp. 499-501.

Opere, VIII, 76

ff.

n

Opere, VIII, 199-200.

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limit concept can precise meaning be given to either the sum of an infinite series or to a first or last ratio. Whatever the influences which shaped the thought of Galileo, those felt by his friend and student, Cavalieri, are not likely to have been greatly different. Although familiar with the views of Valerio and perhaps also with those of Stevin,™ Cavalieri did not develop the limit idea which these men had adumbrated. Instead he had resort to the less subtle notion of the indivisible which had been adopted by Galileo.80 However, whereas Galileo had employed this in physical explanation, Cavalieri made it the basis of a geometrical method of demonstration which achieved remarkable popularity. This method had been developed by Cavalieri as early as 1626, for in that year he wrote to Galileo, saying that he was going to publish a book on the subject. 81 This work appeared in 1635, as the Geometría indivisibilibus continuorum nova quadatn ratione promota,B and the method it presented was further developed a dozen years later in the Exercitationes geometricae sex. Cavalieri at no point in his books explained precisely what he understood by the word indivisible, which he employed to characterize the infinitesimal elements used in his method. He spoke of these in much the same manner as had Galileo in referring to the parallel lines representing velocities or moments as making up the triangle and the quadrilateral. Cavalieri conceived of a surface as made up of an indefinite number of equidistant parallel lines and of a solid as composed of parallel equidistant planes,83 these elements being designated the indivisibles of the surface and of the volume respectively. Although he recognized that the number of these must be indefinitely great, he did not follow his master Galileo in speculations as to the nature of the infinite. The attitude of Cavalieri toward infinity was one of agnosticism.84 He did not share the Aristotelian view of infinity as indicating a potentiality only—a conception which, in conjunction with the work of Stevin and Valerio, pointed toward the method of limits. On the other hand, Cavalieri did not join Nicholas of Cusa and Kepler in ren

Bosmans, "Les Démonstrations par l'analyse infinitésimale," p. 211. Marie, Ilistoire des sciences mathématiques ei physiques, III, 134. 81 Moritz Cantor, Vorlesungen, II, 759. a I have used the later edition, Bononiae, 1653. ° Exercitationes geometricae sex, p. 3. M Cf. Brunschvicg, Les Étapes de la philosophic mathématique, p. 166. 80

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garding the infinite as possessing a metaphysical significance. I t was employed by him solely as an auxiliary notion, comparable to the "sophistic" quantities of Cardan. Inasmuch as it did not appear in the conclusion, its nature need not be made clear. That the infinite did not enter explicitly into the arguments of Cavalieri was due to the fact that at every stage he centered attention upon the correspondence between the indivisibles of two configurations, rather than upon the totality of indivisibles within a single area or volume. The proposition still known in solid-geometry textbooks as Cavalieri's Theorem is characteristic of his approach: If two solids have equal altitudes, and if sections made by planes parallel to the bases and at equal distances from them are always in a given ratio, then the volumes of the solids

Typical of the propositions in the method of indivisibles, and of farreaching significance in later developments, were a number of theorems on the lines of a parallelogram and those of its constituent triangles. One of these propositions consisted in an extended demonstration that if a parallelogram AD (fig. 11) is divided by the diagonal CF into two triangles, ACF and DCF, then the parallelogram is double either triangle. 86 This is proved by showing that if one lay off EF = CB and draw HE and BM parallel to CD, then the lines HE and BM are equal. Therefore all the lines of triangle ACF taken together are equal to all those of CFD; consequently triangles ACF and CDF are equal, and " Geometría indivisibilibus, pp. 113-15; Exercitationes geometricae sex, pp. 4-5. Cf. also Evans, "Cavalieri's Theorem in His Own Words." " Geometría indivisibilibus, Proposition X I X , pp. 146-47; Exercitationes geometricae sex, Proposition X I X , pp. 35-36.

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the sum of the lines of the parallelogram AD is double the sum of the lines of either triangle. From here Cavalieri went on to prove by a similar, but considerably more involved, argument that the sum of the squares of the lines in the parallelogram is three times the sum of the squares of the lines in each of the constituent triangles. 87 Using this latter proposition, he then easily demonstrated, among other things, that the volume of a cone is 3 that of the circumscribed cylinder, and that the area of a parabolic segment is § the area of the circumscribed rectangle. 88 These results were of course known to Archimedes, but a problem which Kepler had proposed some years before now led Cavalieri to a use of indivisibles bolder than that which is found in the Geometria, and to a new result of significance.89 Kepler had, in his Stereometria, challenged geometers to find the volume of the solid obtained by rotating a segment of a parabola about its chord. 90 Cavalieri determined this volume by basing the problem on the discovery that the sum of the fourth powers of the lines of a parallelogram is 5 times the sum of the fourth powers of the lines of one of the constituent triangles. Then he recalled that in his Geometria indivisibilibus he had found the ratios of the lines to be 2 to 1 and the ratio of the squares of the lines to be 3 to 1. In order not to leave a gap in his results on the ratios of the powers of the lines of a parallelogram and of the triangle, Cavalieri sought the ratio for the sums of the cubes of the lines, and found this to be as 4 to 1. He then concluded by analogy that for the fifth powers it would be 6 to 1, for sixth powers 7 to 1, and so on, the sum of the nth powers of the lines of the parallelogram being to the sum of the nth powers of the lines of the triangle as n + 1 to l. 91 The method of Cavalieri here employed was based upon several lemmas which are equivalent to special cases of the binomial theorem. 92 For example, to prove that the sum of the cubes of the lines of a paral87 Geometria indivisibilibus, Proposition XXIV, pp. 159-60; Exercitationes geometrical sex, Proposition X X I V , pp. 50-51. 88 Geometria indivisibilibus, pp. 185, 285-86; Exercitationes geometricae sex, pp. 78 ff. 88 For a summary of this work, see Bosmans, "Un Chapitre de l'œuvre de Cavalieri." M Opera omnia, IV, 601. 91 Exercitationes geometricae sex, pp. 290-91. For a statement of the manner in which he was led to this result, see pp. 243-44. n Exercitationes geometricae sex, p. 267.

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lelogram is four times the sum of the cubes of the lines of one of the constituent triangles, he began with (a + b)3 = a3 + 3a 2 i + Sab2 + b3. Then he proceeded somewhat as follows: setting A F = c, GH = a, HE = b, in figure 11, we have 2c3 = 2a 3 + 32a2è + 3Zai2 + 2b 3 , where the sums are taken over the lines of the parallelogram and triangles. By symmetry this can be written 2c3 = 22a3 + 62a2b. Now 2c3 = c2c2 = c2 (a + ¿>)2 = c2a2 + 2c2a6 + cXb2. But in an earlier proposition on lines Cavalieri had shown that 2a 2 = 2b2 = ^2c2. This gives us 2c3 = fcSc2 + 2c2a& = §2 c3 + 2 (a + b)2ab = 12c3 + 22a26 + 22 b2a = 52c3 + 42 a2b or 2a?b = tjSc 3 . Substituting this in the equation above, we obtain 2c3 = 22a3 + 12c3, or 2c3 = 42a3, and the proposition is proved. 83 Cavalieri realized that this method can be generalized for all values of n, M but he gave complete demonstrations only up to and including n = 4. For higher values he gave only some "cossic" indications, which had been given him by Beaugrand. 96 These may well have been derived from the contemporaneous work of Fermât, 96 whose methods we shall consider later. The results of Cavalieri outlined above are in a general sense equivaÛ- + 1 lent to what we would now express by the notation x"dx = . n +1 We have seen that Archimedes, through the use of series in connection with his stereometric work, had recognized the truth of this proposition for n = 1 and n = 2. He may have known of it for n = 3 also, and the Arabs proved it for n = 4 as well.97 Cavalieri's work, although based upon somewhat different views, was a generalization of these results of Archimedes and the Arabs. It will be seen later that in this respect Cavalieri had been anticipated by several contemporary mathematicians. Nevertheless, his statement, which appeared in 1639,98 represents the first publication of this theorem, which played a signifi» Ibid., pp. 273-74. " "Et sic in infinitum." Ibid., p. 268. M " Ibid., pp. 286-89. See Fermât, Œuvres, Supplement, p. 144. " See Paul Tannery, "Sur le sommation des cubes entiers dans l'antiquité," and Ibn al-IIaitham (Suter), "Die Abhandlung ùber die Ausmessung des Paraboloids." " Centuria di varii problemi, pp. 523-26. Simon ("Zur Geschichte und Philosophie der Differentiairechnung," p. 118) has said—very probably through some error—that Cavalieri found this result in 1615.

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cant rôle in the development of infinitesimal methods during the period from 1636 to 1655. Within this interval, the mathematicians Tonicelli, Roberval, Pascal, Fermât, and Wallis arrived, all more or less independently and by varying methods, at this fundamental result and extended it, as well, to include negative, rational fractional, and even irrational values of n. It was perhaps the first theorem in infinitesimal analysis to point toward the possibility of a more general algebraic rule of procedure, such as that which, formulated a generation later by Newton and Leibniz, became basic in the integral calculus. Cavalieri himself had no vision of such a new analysis; neither he nor Galileo appears to have been seriously interested in algebra, either as a manner of expression or as a form of demonstration. The proposition remained for Cavalieri a geometrical theorem concerning the ratio subsisting between a sum of powers of the lines of a parallelogram and that of one of the constituent triangles. Furthermore, he did not express any clear conception—such as is basic in our idea of the definite integral—of his ratios as limits of sums of infinitesimal parallelograms. Cavalieri never did make clear his interpretation of the indivisible and in this respect laid his method open to attack. Cavalieri's use of indivisibles in his Geometría had been criticized by the Jesuit, Paul Guldin, who asserted not only that the method had been taken from Kepler, but also that it was incorrect, inasmuch as it led to paradoxes and fallacies. Cavalieri defended himself from the first charge by pointing out that his method differed from that of Kepler in that it made use only of indivisibles, whereas Kepler had thought of a solid as made up of very tiny solids." In answering the charge that the method was invalid, Cavalieri maintained that although the indivisibles may correctly be considered as having no thickness, nevertheless, if one wished, he could substitute for them small elements of area and volume in the manner of Archimedes. Guldin had said that since the number of indivisibles was infinite, these could not be compared with one another. Furthermore, he had ** " E x minutissimis corporibus." Exercitationes geometricae sex, p. 181; see also Kepler, Opera omnia, IV, 656-57, for notes, including extracts from Cavalieri on this point. Cavalieri's assertion in this connection should be sufficient to refute the statement often made (see e. g., Wolf, A History of Science, Technology, and Philosophy in the Sixteenth and Seventeenth Centuries, p. 206) t h a t he was probably well aware of the fact that his indivisibles must be of the same dimension as t h a t of the figure which they constitute.

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pointed out a number of fallacies to which the method of indivisibles appeared to lead. In answering these arguments, Cavalieri said that difficulties in the method are avoided by observing that the two infinities of elements to be compared are of the same kind. If, for example, the altitudes of two figures are unequal, their horizontal sections are not to be compared, because the corresponding indivisibles of one are not the same distance apart as are those in the other. Whereas in one figure there may be 100 indivisibles between two sections, between the corresponding sections of the other there may be 200.100 This explanation Cavalieri followed with an ingenuous comparison of the indivisibles of a surface with the threads of a piece of cloth, and those of a solid with the pages of a book.101 Although in geometrical solids and surfaces the indivisibles are infinite in number and lacking in all thickness, nevertheless they may be compared in the same manner as in case of the cloth and the book, if one observes the precaution mentioned above. Cavalieri did not explain how an aggregate of elements without thickness could make up an area or volume, although in a number of places he linked his method of indivisibles with ideas of motion. This association had been suggested somewhat vaguely by Plato and the Scholastic philosophers, and Galileo had followed them in associating dynamics with geometrical representation. Napier, in 1614, had likewise employed the idea of the fluxion of a quantity to picture by means of lines the relation between logarithms and numbers.192 Cavalieri followed this trend in holding that surfaces and volumes could be regarded as generated by the flowing of indivisibles.101 He did not, however, develop this suggestive idea into a geometrical method. This was done by his successor Torricelli, with the result that it eventuated later in Newton's method of fluxions. Cavalieri's indivisible itself was also to find a counterpart in the thought of both Newton and Leibniz—in the former's conception of moments and in the latter's notion of differentials. Cavalieri's vague suggestions were thus to play a large part in the development of the calculus. The Geometría inéivisibilibus of Cavalieri achieved popularity almost immediately, and became, except for the works of Archimedes, 100 101 1-(6)2 + 5(6) The second term on the right in each line is half of the side, and represents the excess of the triangle over the half square. This continues to diminish, in proportion to the first term, as the number of points and lines is increased. Since the number of lines in a geometrical triangle or square is infinite, the excess or half of one line, "does not enter into consideration."174 It is therefore clear that the triangle is.half the square. This argument is evidently roughly equivalent to that india2 cated by the notation f^xdx = —. Roberval continued this type of work by remarking similarly that if the lines should follow one another in the order of the square, the sum of all these little lines, or the points which represent them, would be to the last, taken an equal number of times, as the pyramid is to the prism, that is, as 1 to 3. For example, if we take a square pyramid of dots having four on an edge, we have l 2 + 2- + 32 + 42 = 30 = -J-(4)3 + K4) 2 + i ( 4 ) ; if there are five on an edge, we have l 2 -f- 22 + 32 + 42 + 52 = 55 = 1(5) 3 + §(5)2 + ¿(5); and so on. In these equations the first term on the right is one-third the cube, the second is one-half the square, and the last is one-sixth the number of points in the edge of the base of the pyramid. Inasmuch as the number of squares in a geometrical cube is infinite, the last two terms are as nothing, so iH "Divers ouvrages," pp. 247^18.

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that the sum is J the cube. 17 ' In the same way, the sum of the cubes is one-fourth of the fourth power; the sum of the fourth powers is onefifth of the fifth power; the sum of the fifth powers is one-sixth of the sixth power, and so on.178 In other words, Roberval has in this manner a- + 1 indicated the equivalent of the theorem, ¡"(pc'dx = , for positive n + 1 integral values of n. He appears not to have given a demonstration for other values of m,177 as did Torricelli and Fermat. The arguments of Roberval resemble the "demonstrations by arithmetic" given by Stevin half a century earlier, and similar ones of Wallis a few years after Roberval's work.178 All of these represent efforts to express the notion of limit, but Roberval in his work obscured 1 >C 1 B E' F'• G' H' I'1 y T

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the limit idea somewhat by resorting to his notion of indivisibles. Instead of drawing his conclusion from the limits of the arithmetic sequences involved, he had recourse, as did most of his contemporaries, to geometrical intuition. Making use of the Pythagorean and Nicomachean association of numbers with geometrical points, he remarked that since "the side has no ratio to the cube, . . . adding or subtracting a single square has no effect." 179 Intuition of this type led, 178 Ibid., pp. 248-49. Ibid., p. 248. Zeuthen, "Notes," 1895, p. 43. Walker (A Study of the Traiti des Indivisibles of Roberval, p. 165) makes no mention of Stevin in this respect, incorrectly representing Roberval's work as the first of its type. The statement that the idea of an arithmetic limit appeared for the first time in the seventeenth century (ibid., p. 35) is, of course, not accurate, inasmuch as Stevin's arithmetical work in limits appeared in 1586. "»"Divers ouvrages," p. 249. m

178

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through the neglect of infinitesimals of higher order, to the differential calculus of Leibniz, rather than to the method of limits which Newton suggested and which triumphed in the end. Roberval successfully applied this quasi-arithmetical method of indivisibles to varied problems in quadratures. Typical of these is his quadrature of the parabola.180 The procedure which he employed is quite different from any given by Torricelli in his twenty-one quadratures of the parabola. Roberval's method resembles, rather, Stevin's demonstration by numbers, reinforced by intuition of indivisibles. Let AE = 1, AF = 2, AG = 3, . . . (fig. 17). Then from the definition of EL AE? the Kparabola we know that — = , and similarly J for the other FM AF1 points of division. Hence area ADC _ AE(EL + FM + GN + . . .) area ABCD ~ AD • DC _ AE{AE? + AF2 + AG1 + • . .) AD•AD2 2 2 _ l + 2 + 32 + . . . + AD2 _ j ~ AD • AD1 ~ ~ 5 Roberval by similar methods found the areas under other curves, such as parabolas of higher degrees, the hyperbola, the cycloid, and the sine curve, as well as various volumes and centers of gravity associated with these. He may in this connection have anticipated Torricelli in finding the volumes of infinitely long solids. He also used an ingenious transformation of one figure into another which came to be spoken of as the method of Robervallian lines and which resembled the ductus plani in planum of Gregory of St. Vincent. Such transformations played a large part in the geometry of the seventeenth century because of the lack of a facile method for handling curves whose equations involved radicals, but after the development of the calculus these lost their popularity, as well as their significance. Roberval in his work showed a remarkable flexibility, using divers infinitesimal elements, such as triangles, parallelograms, parallelepipeds, cylinders, and concentric cylindrical shells. Throughout it all, w Ibid., pp. 256-59.

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the idea of limits is implied, but is concealed under the terminology of Roberval's method of indivisibles. An adumbration of the method of limits is indicated also by the manner in which Roberval reconciled the demonstrations by indivisibles with those of the ancient geometers. First Roberval showed that the unknown quantity lies between inscribed and circumscribed figures which differ "by less than every known quantity proposed." Then he showed that the quantity in question bore to the circumscribed figure a ratio less, and to the inscribed figure a ratio more, than the proposed ratio. Finally Roberval proved the proposition by the application of a general lemma: "If there is a true ratio R: S and two quantities A and B such that for a small (quantity) added to A, then this sum has to B a greater ratio then R: S and for a small (quantity) subtracted from A, the remainder has to B a ratio less than R: S; then I say t h a t A : B as R: S."m This form of argument, resembling strikingly the corresponding projxjsitions of Valerio (with whose work he may have been familiar), is equivalent to a statement that the limit of a quotient of variables is equal to the quotient of their respective limits. In Roberval's propositions on indivisibles one rccognizes numerous anticipations of the integral calculus, some of which are equivalent to the determination of definite integrals of algebraic and trignometric functions. Roberval was concerned, as well, with problems of the differential calculus—for he developed a method of tangents so much like t h a t of Torricelli t h a t charges of plagiarism arose.182 He regarded every curve as the path of a moving point, and accepted as an axiom t h a t the direction of motion is also that of the tangent. 183 By looking upon the motion of the point as made up of two component movements, he found the tangent by determining the resultant of these. Thus he found the tangent to the parabola by making use of the fact that, since this curve is the locus of points equidistant from the focus and the directrix, it may be regarded as generated by a point moving with a compound motion made up of a uniform motion of translation away from the directrix and an equal uniform radial motion away from the focus. By the parallelogram (in this case a rhombus) of velocities it is 181

Walker, A Study of the Traite des Indivisibles of Roberval, pp. 38-39. "Divers ouvrages," pp. 436-78; Walker, op. cit., pp. 142-64. Moritz Cantor (Vorlesungen, II, 808-14) concludes that the charges are not substantiated. w "Divers ouvrages," p. 24.

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therefore determined that the resultant velocity—and consequently the tangent to the parabola at any point—will be in the direction of the bisector of the angle between the focal radius at the point and the perpendicular from the point to the directrix. This direction being known, the tangent can be drawn.184 The motions involved here are different from those used for the same curve by Torricelli, but the underlying idea of the composition of movements is essentially the same. The method is, of course, subject to the difficulty that one must in some way discover the laws of motion before one can determine the tangent. Roberval appears, from his correspondence with Fermat in 1636, to have had another method of tangents which proceeded analytically and which he says was connected with the problem of quadratures. 184 This might have been significant in the history of the calculus, but it apparently was lost. I t is difficult to determine the extent of Roberval's influence on contemporary mathematicians inasmuch as his Traite des indivisibles was not published until 1693—that is, only after the calculus itself had been made known. It is very probable, however, that he had a strong influence upon Pascal the younger, whose father, Etienne Pascal, was a close friend of Roberval. Blaise Pascal in a sense represents the highest development of the method of infinitesimals carried out under the traditions of classical geometry. He was not so much a creative genius as a mathematician, scientist, and philosopher, with a remarkable flair for clarifying ideas which had been somewhat vaguely set forth by others, and for supplying these with a reasonable basis.186 This penchant of Pascal's is well illustrated in science by his lucid organization of the principles of hydrostatics; in mathematics one sees it in his exposition of the nature of infinitesimals, in which, to be sure, one perceives also a touch of his characteristic mystical turn. Pascal was not a professional geometer, and as a result his geometrical work was accomplished in two periods which were separated by an interval of mathematical inactivity (from 1654 to 1658) during which he devoted his interests to theology. These two periods, moreover, are characterized by somewhat different views as to the nature 184 u

Ibid., pp. 24-26. See Moritz Cantor, Vorlesungen, II, 812. * Bosmans, "Sur l'œuvre mathématique de Biaise Pascal.'*

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of infinitesimals. Pascal had two predominating interests in mathematics: geometry and the theory of numbers. Toward the end of the first of his two periods of mathematical work the latter was dominant, and at this time he applied the theory of infinitesimals to his work on the arithmetic triangle. Although this is usually called Pascal's triangle, the ordering of binomial coefficients had been known to Stifel long before.187 In this connection he enunciated, in the Potestalum numericarum summa of 1654, the theorem on the integral of x", which we have met with in the work of Cavalieri, Torricelli, and Roberval. Pascal's demonstration of this188 is derived not from classic geometrical propositions alone, but from an examination of the figurate numbers represented in the arithmetic triangle—a form of proof suggestive of that of Roberval, which appears not to have been generally known at Paris at the time. 1 " In the arithmetic triangle the numbers in the first row (or column) may be considered units or points making up a line. Those in the second row represent the sums of the numbers in the first row and may be 1 1 1 1 1 . . 1 2 3 4 . . 1 3 6 . . 1 4 . . 1 . . considered therefore as the sums of points or units—that is, as lines. The numbers in the third row, which are in turn the sums of those in the second row, may therefore be considered as the sums of lines, that is, as triangles. The numbers in the fourth row similarly represent pyramids. Geometrical intuition now fails, but one can continue by analogy. 190 From such geometrical considerations and from the numerical relationships within the triangle, Pascal was led immediately to consider the sums of powers of the positive integers. Recalling the results by the geometrical procedures of the ancients for the sums of the 187

See Bosmans, "Note historique sur le triangle arithmétique de Pascal." M Œuvres, III, 346-67, 433-593. Zeuthen, "Notes," 1895, p. 43. m Cf. Bosmans, "Sur l'interprétation géométrique, donnée par Pascal à l'espace à quatre dimensions." m

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squares and of the cubes, he recognized that these were not immediately applicable to powers of higher degree. Pascal, however, developed a general arithmetical method for determining the sum, not only in the case of terms which are integral powers (of the same degree) of the first N natural numbers, but also for powers (of the same degree) of any integers in arithmetic progression. Pascal expressed his result rhetorically, but this may be given in symbolic form by the equation : H+ 'C1dZ(') + " + 1 C 2 d 2 S ( " _1) + . . . + " + IC„¿"2(1) = (a + Nd)H + l - an + i - Nd" + where a is the first term of the progression, d the common difference, N the number of terms, n the degree of the powers in question, " + 'C,the number in the (i + l)st column and the (n — i + 2)nd row in Pascal's triangle, and 2 o ' the sum of thejlh powers of the terms of the progression. As Pascal remarked, it will be obvious to any one who is at all familiar with the doctrine of indivisibles that this result can be applied to the determination of curvilinear areas. In order to find the area under the curve y = x", for example, the surface in question is to be regarded as the sum of ordinates which are the nth powers of abscissae chosen in arithmetic progression (with first term zero and with common difference equal to unity), of which in this case there will be an infinite number. Moreover, a single point adds nothing to the length of a line : nor does the addition of a line to a surface cause any difference in area, for the former is an indivisible with respect to the latter. Or, speaking arithmetically, roots do not figure in a ratio of squares, nor squares in a ratio of cubes, and so on.181 The rule given above consequently becomes, on calling the greatest abscissa b and on neglecting as zero the terms of lower order, (n -f- l)s("} = b" + In a general sense this is, b" + 1 of course, the equivalent of the expression J"0xndx = . n +1 Or, translating this into the terminology often used at the time, the sum of lines in a triangle is half the square of the longest; the sum of the squares of the lines is one-third the cube ; the sum of the cubes is onefourth the "square-square," and so on. The essential point in Pascal's demonstration is the omission of m Œuvres, HI, 366-67.

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terms of lower dimension. This type of argument has frequently been attributed to Cavalieri,192 but there appears to be no basis for such a view. Cavalieri's method was based upon a strict correspondence of the indivisibles in two figures, and there were no unpaired or omitted elements. The method of dropping terms seems to have entered in the work of Roberval and Pascal through the association of the indivisible of geometry with arithmetic and the theory of numbers. The geometrical intuition of indivisibles of lower dimension was, in their work, carried over into arithmetic to justify the neglect of certain terms of lower degree. Pascal went so far as to compare the indivisible of geometry with the zero of arithmetic, much as Euler later regarded the differentials of the calculus as nothing but zeros. This neglect of quantities, as found in Pascal, has been characterized193 as the basic principle of the differential calculus. Such a designation is indeed misleading, for the subject is no longer explained in terms of the omission of fixed infinitesimals. Nevertheless, the work of Pascal exerted perhaps the strongest influence in shaping the views of Leibniz, who adopted into his calculus as fundamental the doctrine that "differences" of higher order could be neglectcd. Newton also occasionally lapsed into this type of argument in dropping out of the calculation "moments" which did not add significantly to the result. For almost two centuries mathematicians tried to justify such procedures, but in the end the basis of analysis was found, not in these, but rather in the method of limits toward which the geometrical method of exhaustion and arithmetical modifications of this by Stevin, Tacquet, Roberval, and others had pointed. In answering the objections of those of his contemporaries who held that the omission of infinitely small quantities constituted a violation of common sense, Pascal had recourse to a favorite theme—that the heart intervenes to make this work clear. In this case what is necessary is the "esprit de finesse," or intuition, rather than the "esprit de géométrie," or logical thought, much as the action of grace, as well as physical experience, is above reason. In this respect the paradoxes of geometry are to be compared to the apparent absurdities of Christiln Ball, History of Mathematics, p. 249; Cajori, History of Mathematics, p. 161; Marie, Histoire des sciences mathémaliques, IV, 72; Milhaud, "Note sur les origines du calcul infinitésimal," p. 35. 183 Simon, "Zur Geschichte und Philosophic der Differentialrechnung," p. 120.

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anity, the indivisible being to geometrical configurations as our justice to God's.1»4 The mysticism which Pascal often displayed in his attitude toward the infinitesimal does not appear in all of his work. Particularly in the later period of his mathematical activity—in which his interest centered about the cycloid, the curve which Montucla called "la pomme de discorde"195 because it engendered so many quarrels with respect to priority—his view appears to be modified. In connection with problems such as those in his Traité des sinus du quart de cercle of 1659 in which he balanced elements as Archimedes had done in his mechanical method, he used the language of infinitesimals in speaking of the sum of all the ordinates; but he added that one need not fear to do this, inasmuch as what is really meant is the sum of arbitrarily small rectangles.196 In his later numerical demonstrations also Pascal sought to avoid arguments based upon the neglect of infinitely small quantities. The Aristotelian view had denied the existence in the realm of number of the infinitely small, while admitting, as a potentiality, the infinitely great. Pascal, on the contrary, maintained in De l'esprit géométrique that in the sphere of number the infinitely great and small are complementary. Corresponding to every large number, such as 100,000, there existed a small one, the reciprocal 1 0 0 ' 0 0 0 so that the existence of the indefinitely large implied that of the indefinitely small. Number, he held, was as much subject to the two infinities—in greatness and in smallness—as were such other undefined primitive terms in geometry as time, motion, and space.197 The contrast between discrete and continuous magnitudes was not so great as Aristole had felt, and was, in fact, vanishing with the spread of analytic methods in geometry. The change in Pascal to a clear point of view with respect to infinitesimals may have been the result of friendship with Roberval, who had said that Cavalieri did not really think of indivisibles as lines. It may equally well have come from Pascal's reading of Tacquet's Cylindricorum et annularium,l%% in which the author denied the validity of concluding anything about the ratio of surfaces from the ratio of their indivisibles, or lines. Tacquet had been particularly emphatic in denySee Œuvres, XII, 9, XIII, 141-55. Montucla, Histoire des mathématiques, II, 52. w Œuvres, IX, 60-76. Œuvres, IX, 247, 253, 256, 268. 158 Bosmans, "La Notion des indivisibles chez Blaise Pascal."

195

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ing that a configuration could be thought of as composed of heterogenea, or elements of a lower dimensionality. Pascal was in general agreement with him on the question of homogeneity, but his view of the transition from the finite to the infinite was different. Tacquet was inclined toward the limit idea of Gregory of St. Vincent, although he preferred to avoid the difficulty by returning to the clarity afforded by the method of exhaustion. Pascal, on the other hand, looked upon the infinitely large and the infinitely small as mysteries—something which nature has proposed to man, not to understand, but to admire. 19 ' Furthermore, Tacquet had made use of the limits of infinite series, as had also Stevin and Roberval. The work of Pascal, however, was carried out in connection with the older theory of numbers and classical geometry of which he considered his method an elaboration. The newer analytic procedures of Fermât and Descartes did not appeal to him, and for them he substituted a remarkable facility in the manipulation of geometric transformations, similar to those of Gregory of St. Vincent and Roberval. Through these he related the figúrate sums of his number theory to problems in the synthetic geometry of continuous magnitudes and anticipated numerous results of the integral calculus, including the equivalent of integration by parts. His underestimation of the value of the algebraic and the analytic viewpoints may have been responsible not only for his inability to define the central and unifying concept of the integral calculus—that of a limit of a sum—but also for his failure to recognize the inverse nature of the problems of quadratures and tangents. The idea and figure of what is now called the differential triangle had appeared on several occasions before the time of Pascal, and even as early as 1624. Snell, in his Tiphys Batavus, had thought of a small spherical surface bounded by a loxodrome, a circle of latitude, and a meridian of longitude as equivalent to a plane right triangle. 200 Numerous diagrams somewhat resembling the differential triangle are to be found in the geometrical works of the middle seventeenth century, such as those in the De infmitis hyperbolis of Torricelli and the Traité des indivisibles of Roberval, with which Pascal may have been familiar. m

Œuvres, IX, 268. Aubry, "Sur l'histoire du calcul infinitésimal entre les années 1620 et 1660," p. 84.

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In all of these, however, the significance of the quotient of two sides of the triangle for the determination of tangents appears to have escaped emphasis. It was much the same with Pascal. In connection with a diagram (see fig. 18) from his Traité des sinus du quart de cercle of 1659, he remarked that A D is to DI as EE is to RR or EK, and that for small intervals the arc may be substituted for the tangent. Pascal made use of these lemmas to determine the sum of the sines (ordinates) of a portion of the curve, that is, the area under this portion. If Pascal had at this point only been more interested in arithmetic considerations and in the problem of tangents, he might have anticipated the important concept of the limit of a quotient and have discovered the

c

R

I

R

A

FIGURE 18 significance of this for the determination of both tangents and quadratures. Had he done this, he would have hit upon the crucial point in the calculus some seven years before Newton and about fourteen years before Leibniz. The latter, who later, as we shall see, made use of this very diagram to establish his infinitesimal calculus, said, in 1703, in a letter to James Bernoulli, that sometimes Pascal seemed to have had a bandage over his eyes.201 This apparent lack of imagination was very likely the result of a predilection for the classical, such as later restrained the scientist Huygens also from making full use of the new procedures. 101

See Leibniz, The Early I I I , 72-73, n.

Sckriften,

Mathematical Manuscripts,

pp. 15-16, and

Mathematische

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Pierre de Fermât, the friend of Pascal and perhaps the greatest French mathematician of the century, possessed a singular erudition and displayed an enthusiastic interest in Greek and Latin philology. This led him to study carefully such classic mathematical works as those of Archimedes, Apollonius, and Diophantus. The influence of the first of these three had been very strong for almost two centuries, but in the work of Viète and of Fermât we have evidence of the impression made by the other two also, as well as by the Arabic and Italian development of algebra. Viète realized the facility to be gained in the handling of geometric problems by their reduction to the solution of algebraic equations, a procedure which he therefore followed whenever possible.202 Viète's equations betrayed their origin in geometry, in that he was always careful to have them all homogeneous; but his work was nevertheless, in a sense, an inversion of the Greek view, in accordance with which algebraic equations were reduced to geometric constructions for purposes of solution. Fermât was familiar with the methods of Viète and developed them into an analytic geometry at about the time that Descartes was preparing his famous Géométrie of 1637. The work of Fermât and Descartes went much further than did either the algebraic solution of geometric problems by Viète or the graphical representation of variables by Oresme, for it associated with each curve an equation in which are implied all of the properties of the curve. This recognition, which Fermât expressed in calling the equation the "specific property" of the curve, constitutes the basic discovery of analytic geometry. Although in publication Fermât was anticipated by Descartes, he far outdid his rival in the application of the new point of view to the problems of infinitesimal analysis, which the books of Kepler and Cavalieri had popularized. All of the anticipations of methods of the calculus which we have so far considered were related to geometry. Infinite series had sometimes been employed, but they were derived from the geometrical representation of the problem. Infinitesimal lines, surfaces, and solids had been used, but not infinitesimal numbers. Aristotle had denied the infinitely small in arithmetic, for the obvious reason that since number was a 502 For an unusually extensive account of this work, see Marie, Histoire des sciences mathématiques, III, 27-65.

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collection of unities, no number could be smaller than one. As a result of the algebra and the analytic geometry of the sixteenth and seventeenth centuries, this attitude had been modified, as has already been seen in the case of Pascal. The present-day view of the symbols entering into an equation is that they represent, in general, continuous variables; for Fermât and Descartes, however, they represented indeterminate constants 203 to which line segments could be associated, the tacit assumption being made that to every segment there corresponded some number. To such a view there was nothing incongruous in the idea of infinitesimal constants or numbers, since they would correspond to the geometrical infinitesimals which were being used so successfully. These numerical infinitesimals arose first through some interesting problems considered by Fermât. Pappus had spoken of a "minima et singularis proportio," and this led Fermât to dwell on the fact, as he explained in a letter of 1643, that in a problem which in general has two solutions, the maximum or minimum value gives but a single solution.2M Thus if a line of length a is divided by a point P into two parts, x and a — x, there are in general two positions of P which will make the area of the rectangle on x and a — x a given quantity, A. For the maximum area, however, there is only one position, the mid-point. From this fact Fermât was led to formulate his remarkably ingenious and fruitful method for determining maximum and minimum values. His method first appeared in an article in 1638; but Fermât said that the discovery went back some eight or ten years previous.206 The argument in this is as follows: Given a line segment of length a, mark of! from one end a distance x. The area on the segments x and a — x will then be A = x(a — x). If instead of the distance x one were to mark off the distance x + E, the area would be A = (x + E) (a — x — E). For the maximum area the two values will be the same, from Pappus' observation, and the points x and x + E will coincide. Consequently, 201 Wallner, "Entwickelungsgeschichtliche Momente bei Entstehung der Infinitesimalrechnung," p. 119. 204 See Giovannozzi, "Pierre Fermât. Una lettera inédita." 205 See Wieleitner, "Bermerkungen zu Fermats Methode der Aufsuchung von Extremenwerten." In a letter to Roberval, written in 1636, Fermât said that in 1629 he was in possession of his method of maxima and minima. See Paul Tannery, "Sur la date des principales découvertes de Fermât"; cf. also Henry, "Recherches sur les manuscrits de Pierre de Fermât."

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setting the two values of A equal to each other and letting E vanish, a the result is x = -.206 2

The procedure which Fermât here employed is almost precisely that now given in the differential calculus, except that the symbol Ax (or occasionally h) is substituted for E. In his work there appeared, for perhaps the first time, the idea which has become basic in such problems—that of changing the variable slightly and then letting this change vanish. However, the reasoning by which Fermât supported his method is far less clear than that given at the present time. Modern analysis makes use of the concept of the limit, as the change Ax approaches zero. Fermât, however, seems to have interpreted the operation as one in which E vanishes in the sense of actually being zero. For this reason, as Berkeley remarked in the following century, it is difficult to see by what right he took the positions x and x + E to be different and yet in the end said that they coincide. Fermat's argument has frequently been interpreted in terms of the limit concept,207 in which E is to be regarded as a variable quantity approaching zero. Fermât, however, does not appear to have thought of it in this way.208 In fact the function concept and the idea of symbols as representing variables does not seem to enter into the work of any mathematician of the time. In answering criticisms of his method, Fermât presented a statement of his reasoning which appears to link it with the remarks of Oresme and Kepler on the change at a maximum point.209 He justified the equating of the two values of A by remarking that at a maximum point they are not really equal but they should be equal. He therefore formed the pseudo-equality210 which became equality on letting E be zero. From this it is clear that he was thinking in terms of equations and the infinitely small, rather than of functions and the limit concept. 206 Fermât, Œuvres, I, 133-34, 147-51; i n , 121-22; Supplement, pp. 120-25. Cf. also Voss, "Calcul différentiel," p. 246. Duhamel, "Mémoire sur la méthode des maxima et minima de Fermât, et sur les méthodes des tangentes de Fermât et Descartes"; cf. also Mansion, "Méthode, dite de Fermât, pour la recherche des maxima et minima." "•See Wallner, "Entwickelungsgeschichtliche Momente," pp. 122-23; cf. also Paul Tannery, Notions historiques, p. 344. Paul Tannery, however, thinks Fermât borrowed nothing from Kepler, whose works he probably had not read. See his Review of Vivanti, II concetto d'infinitesimo, p. 232. »« "Adaequalitas." Sçç Œuvres, I, 133-79, for his justification.

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None the less, the method worked so beautifully that it found a ready acceptance among mathematicians. As a result, infinitesimals were uncritically introduced into analysis, to become firmly intrenched as the basis of the subject for about two centuries before giving way, as the fundamental concept of the calculus, to the rigorously defined notion of the derivative. Even now the subject is generally known as the "infinitesimal calculus," in spite of the fact that the infinitesimal, while of great pragmatic value in adding to the facility of manipulation of the subject in exercises, is logically secondary and even unnecessary. Fermat was led by the success of his method to apply it, about 1636, to the determination of tangents to curves. This he did as follows: Let the curve be a parabola (fig. 19). Then from the "specific property" of

0

FIGURE 19 the curve it is clear that if we set OQ = a, VQ — b, and QQ' = E, we b a2 shall have > . Torricelli, in his work on the tangents to b+ E (a + E)2 parabolas, had frequently set down such inequalities.211 However, whereas Torricelli had made use of arguments by a reductio ad absurdum, Fermat's characteristic procedure resembles more closely the method of limiting values. Inasmuch as for small values of E the point P' may be regarded as practically on the curve as well as on the tangent line, the inequality becomes, as in the method for maximum values, a pseudo-equality. By allowing E to vanish, this pseudoequality becomes a true equality and gives the desired result, a = 2b.™ m

See Opere, I (Part 2), 304ff.,315 fit.

*> (Euvres, I, 134-36; i n , 122-23.

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The method of tangents Fermat believed to be an application of his method for maxima, but he was unable to explain what quantity he was maximizing. Descartes naturally supposed that it was the length of the line from the curve to a fixed point 0 on the axis of the parabola. However, on carrying out Fermat's method on this assumption, the result he obtained was, of course, different from that of Fermat. What Descartes had really found was the normal to the curve—that is, the minimum distance from a point on the axis to the curve. This would have furnished an excellent illustration of the method for determining a maximum or a minimum value, but inasmuch as Fermat had not given a rule for distinguishing maxima from minima, neither he nor Descartes recognized it as such. Descartes simply concluded that although the result Fermat had obtained was correct, the method was not generally applicable. Perhaps as the result of the unnecessarily bitter criticism and supercilious attitude of Descartes, Fermat later modified his explanation for determining tangents.213 Instead of interpreting the method in terms of maxima, he said that the point P' was indifferently taken as on the curve or on the tangent line. Then after forming the pseudo-equality, the quantity E was to vanish, to give the desired result. This procedure is strictly comparable to that now employed in the calculus, the theoretical justification of which is given in terms of limits; but the explanation of Fermat resembles rather the neglect of infinitesimals to be found in the work of Leibniz. It is also strikingly suggestive of the doctrine of perfect and imperfect equations, presented almost two hundred years later by Carnot in his attempted concordance of the conflicting views of the calculus then prevailing. Fermat applied analogous considerations to the problem of determining the center of gravity of a segment of a paraboloid, again under the misapprehension that he was employing the method of maxima and minima. In this he let the center of gravity, O, of the segment be a units from the vertex. On decreasing the altitude, h, of the segment by E, the center of gravity is changed. Fermat, however, knew from a number of lemmas that the distances of the centers of gravity of the two segments are proportional to the altitudes and that the volumes of the segments are to each other as the squares of the altitudes. By m

Duhamel, "Mémoire sur la méthode des maxima et minima de Fermat," pp. 310-16.

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159

taking moments about O, he was able to make use of these facts to set up a pseudo-equality involving a, h, and E. In accordance with his general principle, he allowed E to vanish and obtained the result a = p.2" The determination of the center of gravity of the paraboloidal segment did not constitute a new result. Archimedes had calculated this some nineteen hundred years earlier in the Method, and Commandino and Maurolycus had rediscovered it only a century before. Nevertheless, this exercise of Fermât is significant in the history of the calculus as the first determination of the center of gravity by means of methods equivalent to those of the differential calculus, instead of by means of a summation resembling those of the integral calculus. Fermat's friend Roberval was astonished that one should be able to obtain by means of this maximum and minimum method a result which had generally been derived from summation considerations. The integral calculus is, of course, implied in the lemmas which Fermât employed in this connection, and the method of maxima served somewhat indirectly to determine simply the value of the constant of proportionality entering into these. Nevertheless, this theorem might have led Fermât to a recognition of the significance of the inverse nature of summation and tangent problems. That this escaped him is the more strange, in that he developed remarkable procedures for the determination of quadratures as well as for tangents. an + 1 an The equivalents of what we express as ¡¿x^-dx = had appeared n +1 in various forms in the work of Cavalieri, Torricelli, Roberval, and Pascal. Fermât also gave demonstrations of this rule—in fact, he may well have anticipated all the others in this respect—one of which is strikingly different from those given earlier.216 In his earlier investigations, about 1636, he appears m to+have 1 made use of the inequalities

1" + 2m + 3" + . . . + ram > > I™ + 2m + 3" + . . . + (n - 1)" m + 1 to establish the result for positive integral values of n. This constituted a generalization of the inequalities of Archimedes which was known also »» Œuvres, I, 136-39; III, 124-26. ,u For an excellent exposition of this method of quadratures, see Zeuthen, "Notes," 1895, pp. 37-80.

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to Roberval. Fermat may also have given a proof, based upon the formation of figurate numbers, similar to that of Pascal.216 However, before 1644 he had found the quadratures, cubatures, and centers of gravity of the "parabolas" of fractional degree amyn = 6V, 2 1 7 curves which he seems to have been the first to propose, but which were investigated also by Cavalieri, Torricelli, Roberval, and Pascal. Fermat, therefore, probably possessed as early as that date a general proof of the theorem for rational fractional values as well, although the definitive redaction of this was not made until 1657.218 In this connection, Fermat's procedure constitutes a generalization of one found in the Opus geometricum of Gregory of St. Vincent, although Fermat may not have known of this work, for he here mentions only Archimedes. Gregory had shown that if along the horizontal asymptote of a rectangular hyperbola points are marked off whose distances from the center are in continued proportion, and if at these points ordinates are erected to the curve, then the areas intercepted between these are equal.21® Fermat modified this process in such a way that it could be applied to both the general fractional hyperbolas and P

parabolas. In finding the area under y = 2®,220 for example, from 0 to x, he would take points on the axis with abscissas x, ex, e2x, . . . where e < 1 (fig. 20). Then erecting the ordinates at these points, the areas of the rectangles constructed upon successive ordinates will form an infinite geometric progression. As Gregory of St. Vincent and Tacquet had earlier found the sums of such progressions, so here Fermat determined the sum of the rectangles to find the area under the curve, however, one must have not only an infinite number of such rectangles, but the area of each must be infinitely small. This can be brought about by setting e = 1. Before Ibid., pp. 42-43. Mersenne, Cogitata physico-mathematica; Œuvres de Fermat, I, 195-98. 118 Zeuthen, "Notes," 1895, pp. 44 ff. n ® Opus geometricum, Proposition CIX, p. 180 The notation of Fertnat has been here clear. The equations of Fermat retained the a7

see preface to Tractalus mechanitus. See also 586. slightly modified to make the meaning more homogeneous character found in Viète.

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161

doing this, however, Fermat made the substitution e = E? in order to evaluate the indeterminate form. The sum then becomes t±*(l ^

\ \ l - E *

+

7

'-Y1

=

*

(1 (1 -

£ ) ( ! +E £)(1

+ &

+ E + E

2

+

. . . +

& ' * )

+

. . . + E,

When e "vanishes," E does likewise, and the sum is then

+

"-i) t±J ax *

,

P+q

which is the area under the curve. By taking e > 1, Fermat applied the same method to the fractional hyperbolas also, finding the area under these from any abscissa to infinity. 221

In these quadratures we see most of the essential aspects of the definite integral—the division of the area under the curve into small elements of area, the approximate numerical determination of the sum of these by means of rectangles and the analytic equation of the curve, and finally an a t t e m p t by Fermat to express the equivalent of what we would call the limit of this sum, as the number of elements is indefinitely increased and as the area of each becomes indefinitely small. One is almost tempted to say that Fermat recognized all the aspects except that of the integral itself; that is, he did not recognize the operation involved as significant in itself. The procedure was for him, as « Œuvres, I, 255-88; III, 216-40.

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it had been for all of his predecessors, simply that of finding a quadrature—of answering a specific geometrical question. Only with Newton and Leibniz were the processes involved in infinitesimal considerations recognized as constituting operations, independent of any geometrical or physical considerations, to which characteristic names were applied.222 That a curvilinear area, such as those under Fermat's parabolas and hyperbolas, could be equal to one bounded only by straight lines had been known in antiquity. It had long been held impossible, however, that a curved line could be exactly equal in length to a straight line,223 and Fermat shared this view with a number of his contemporaries. Sluse and Pascal in this connection expressed admiration for the order of nature, which refused to allow a curve to equal a line.224 Gregory of St. Vincent, Torricelli, Roberval, and Pascal, nevertheless, had by infinitesimal and kinematic means compared the arcs of spirals with those of parabolas. Then, shortly before 1660, there suddenly appeared a number of rectifications of curved lines by William Neil, Christopher Wren, Heinrich van Heuraet, John Wallis, and others.226 These were, in general, based upon approximations to the curve by means of polygons, followed by applications of infinitesimal or limit considerations. Upon hearing of them, Fermat himself carried out a rectification of the semicubical parabola. His procedure in this connection is typical of his general approach and indicates well the interrelation of the various aspects of his work. For any point P on the curve with abscissa OQ = a and ordinate PQ = b, the subtangent, TQ = c, is known by his tangent method to be c = f a (fig. 21). If then an ordinate P'Q' to the tangent line is erected at a distance E from the ordinate PQ, the segment PP' is known in terms of a and E. For the curve ky2 = s 3 , this is PP'

= E

+ 1. But the point

m Simon appears to be overenthusiastic in saying of this work ("Geschichte und Philosophie der Differentialrechnung," p. 119): "F. hatte auch bereits in ähnlicher Weise wie später Riemann den Begriff des bestimmten Integrals erfasst bei der Berechnung von

fx,dx.

0 Hier ist Grenzübergang, hier ist Bestimmung des Werthes - , hier is völliges BeÖ' wusstsein des continuitäts gesetzes." m Kaestner, Geschichte der Mathematik, I, 498; III, 283. *» See Zeuthen, "Notes," 1895, pp. 73-76; and Pascal, Œuvres, VIII, 145, IX, 201. m Moritz Cantor, Vorlesungen, II, 827 ff.

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P', for small values of E, may be regarded as on the curve as well as on the tangent line, so that the length of the curve can be thought of as the sum of segments such as PP'. The sum of these segments, in 9x turn, can be taken as the area under the parabola y2 = f- 1. Inas4k much as the quadrature of this is known, the length of the curve is determined.226 It is surprising that Fermat, who used his method of maxima and minima for finding centers of gravity, who reduced a problem of rectification which involved tangents to a question of quadratures, who

used infinitesimals geometrically and analytically in such a wide variety of problems, should have missed seeing, as Pascal had also, the fundamental connection between the two types of questions. Because these two men did not see this, in problems in which integrations by parts would now be employed they had recourse to clever geometrical transformations. Fermat, in his problems, made use of diagrams which are much the same as the one of Pascal's which Leibniz later found so suggestive of his differential triangle, and yet he did not perceive their deeper significance. Had Fermat only observed more »« For Fermat's rectification, see Œuvres, I, 211-54; HI, 181-215.

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closely the results for the tangents and the quadratures of his parabolas and hyperbolas, he might have discovered the fundamental theorem of the calculus and have become, what he has sometimes been unwarrantedly called, the "true inventor of the calculus."227 Fermat, of course, realized in a sense that the two types of problems had an inverse relationship. That he did not pursue this thought further may well have been due to the fact that he thought of his work simply as the solution of geometrical problems and not as representing a type of argument significant in itself. His methods of maxima and minima, of tangents, and of quadratures he regarded as constituting characteristic approaches to these questions, rather than a new type of analysis. Furthermore, they were apparently restricted in application. Fermat knew how to make use of them only in the case of rational expressions, whereas Newton and Leibniz, through their application of infinite series, recognized the universality of such procedures. Nevertheless, no mathematician, with the possible exception of Barrow, so nearly anticipated the invention of the calculus as did Fermat. The influence of Fermat on his contemporaries and immediate successors228 is difficult to determine. Probably his work was not so well known as that of Cavalieri, for the latter was widely read in his two famous books, whereas Fermat did not publish either his methods or his results. For this failure to publish his work, Fermat, it has been said,22® lost the credit for the invention of the calculus; but such an assertion is incorrect. In the first place, it is clear that he cannot be thought of as its inventor. Secondly, his work was collected and printed posthumously as the Opera varia, in 1679, before either of the earliest published accounts of the calculus by Newton or Leibniz had appeared. In spite of the fact that Fermat did not himself publish his methods, m Lagrange, Laplace, and Fourier have so called him, but Poisson has correctly pointed out that Fermat does not deserve such a designation, inasmuch as he failed to recognize the problem of quadratures as the inverse of that of tangents. The relevant statements of these four men may all be found in Cajori, "Who Was the First Inventor of the Calculus?" Cf. also Marie, Hisioire des sciences mathSmatiques, IV, 93 ff. Sloman has very unfairly said that "Fermat hardly deserves to be named at all" in this connection. See his Claim of Leibnitz to the Invention of the Differential Calculus, pp. 45-47. m See Genty, L'Influence de Fermat sur son siicle. This book is more concerned, however, to show the priority and independence of Fermat's results than specifically to point out their effects on others. m Dantzig, Number, the Language of Science, pp. 131-32.

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these became known through his correspondence with such men as Roberval, Pascal, and Mersenne, as well as through the publication by others during his lifetime of portions of it. As a consequence, his work was in large part responsible for a number of transition methods which appeared just before the advent of the calculus. It is therefore scarcelycorrect to say, with Lagrange,250 that "Fermat's contemporaries did not seize the spirit of this new type of calculus." Infinitesimal considerations such as those of Fermât constituted a large portion of the mathematical activity of the period. Nevertheless, there was indeed one great mathematician who remained somewhat cool toward the new views, even though in his early years he had made effective use of them. This was René Descartes, the severest critic of Fermât. The very first mathematical production of Descartes was an attempt, in 1618, to deal with the laws of falling bodies by means of infinitesimals. In this he made an error, in that he assumed, as Galileo had also in 1604,231 that the velocity was proportional to the distance rather than the time; but if the distance axis in his demonstration were changed to the time axis, his procedure would be essentially that which Oresme and other Scholastics had used.232 Descartes was probably familiar with their works and may have derived this form of proof, as well as suggestions on analytic geometry, from reading Oresme.235 At any rate, Descartes was acquainted with ancient, medieval, and modern views on infinitesimals, and used them. In a second memoir of about the same time he wrote on fluid pressure. In this connection he may have known through Beekman of Stevin's work on infinitesimals. At all events, in considering the force drawing a body, he used such phrases as the "first instant of its movement," and the "first imaginable speed."234 Some years later, in 1632, he answered correctly a number of See Brass ine, Précis des œuvres mathématiques de P. Fermât, p. 4. Duhem, Études sur Léonard de Vinci, HT, 564. m See Descartes, Œuvres, X, 219; cf. also X, 59, 76-77. m There is a great difference of opinion on this subject. Wallner ("Entwickelungsgeschichtliche Momente," p. 120) sees not the least influence of Oresme on Descartes; Stamm ("Tractatus de continuo," p. 24) says that the problem of the latitude of forms of Oresme was undoubtedly the most important influence on Descartes; Wieleitner ("Ueber den Funktionsbegriff," p. 242) says that Descartes undoubtedly knew of Oresme's work, but that the essential idea of the dependence of quantities found in analytic geometry is missing in Oresme; Duhem (Études sur Léonard de Vinci, III, 386) has claimed that Oresme created analytic geometry. Milhaud, Descartes savant, pp. 162-63. m

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questions, which Mersenne had sent him, on areas, volumes, a n d centers of gravity connected with the parabolas y* = px—problems similar to those which F e r m â t had solved. Descartes did not tell w h a t method he had used, b u t it was probably a skillful application of t h e methods of Archimedes, Kepler, and Cavalieri. 234 However, after the publication of his famous Géométrie in 1637, Descartes' interest in t h e subject began to wane, 236 for his mathematical work was only an episode in the development of his philosophy. Consequently he did not participate effectively in the development of the infinitesimal methods which preoccupied the minds of most mathematicians a t this time. 237 T h e acrimonius quarrel with Fermât, however, sustained his interest in the problem of tangents and led him to considerations which, if pursued further, might have been more effective t h a n the infinitesimal methods in leading to a clearer understanding of the basis of the calculus. T h e ancients, with t h e possible exception of Archimedes, had not developed a general definition of a tangent to a curve nor any method of determining it. Descartes, however, realized more fully t h a n a number of his contemporaries t h a t this constituted not only " t h e most useful and general problem t h a t I know b u t even t h a t I have ever desired to know in geometry." 2 3 8 H e thereupon elaborated his celebrated method of tangents in terms of the equality of roots. Descartes' method consisted in passing through two points of t h e curve a circle with its center on the x axis, and then making the points of intersection coincide. T h e center of the circle t h u s becomes the point on the x axis through which the normal to t h e curve passes, and the tangent is consequently known. This m a y be illustrated in somew h a t simplified form as follows: Let the tangent at the point (a, a) on t h e parabola y* = ax be required. T h e equation of the circle going though (a,a) and having its center on the axis is x2 + y" — 2hx + 2ah — 2a2 = 0, where h is undetermined. Substituting ax for y 2 , the quantity h may then be so determined t h a t the resulting equation has equal roots—that is, so t h a t t h e intersections of the circle with the parabola M

Ibid., pp. 164-68. Ibid., p. 246; cf. also Marie, Histoire des sciences mathématiques et physiques, IV, 21. m Milhaud (Descartes savant, pp. 162-63) denies that it was the insistence on clear ideas which made Descartes avoid the use of infinitesimals. «" Descartes, Œuvres, VI, 413.

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9

coincide." This value, h = la, is the abscissa of the point on the axis through which the normal to the parabola passes. The tangent is then the line through (a, a) perpendicular to this normal. It is to be remarked that Descartes' method is purely algebraic, no concepts of limits or infinitesimals being manifestly involved. However, any attempt to interpret geometrically the significance of the case in which the roots are equal, to explain what is meant by speaking of coincident points, or to define the tangent to a curve, would necessarily lead to these conceptions. If Descartes in his geometry had thought in terms of continuous variables rather than of a correspondence between symbols which represented lines in a geometrical diagram,240 he might have been led to interpret his tangent method in terms of limits, and so have given a different direction to the anticipations of the calculus. His algebra, however, was still grounded in the geometry of lines, and the idea of a continuously varying quantity was not really established in analysis until the time of Euler.241 In criticising Fermat's method of tangents, Descartes attempted to correct the method by interpreting it in terms of equal roots and coincident points, a procedure which was practically equivalent to defining the tangent as the limit of a secant.242 Descartes did not express himself in this manner, however, inasmuch as the concept of a limit was far from clear at this time. Fermât, who was thinking of infinitesimals, could not see that his method had anything in common with the algebraic (limit) method of Descartes and so precipitated a quarrel as to priority, one of the many which the seventeenth century produced as the result of the confusion of thought as to the basis of infinitesimal methods. Descartes preferred his method of tangents to that of Fermât because of the apparent freedom from the concept of infinitesimals, although its application was frequently much more tedious and was limited to algebraic curves. In finding the tangent to a nonalgebraic or "mechanical" curve, such as the cycloid, Descartes in 1638 made use of the concept of an instantaneous center of rotation. This is, of course, also directly connected with the use of limits and infinitesimals, but is expressible m

Ibid., VI, 413-24; cf. also Voss, "Calcul différentiel," p. 244. Descartes, Œuvres, VI, 369; cf. also VI, 411-12. 141 Fine, Number-System, p. 121. »» See Duhamel, "Mémoire," pp. 298-308; Milhaud, Descartes savant, pp. 159-62.

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without the use of such terminology, through the circumlocution afforded by the notion of instantaneous velocity. Supposedly, this notion is intuitively clear, but at the time it was not rigorously defined. It had been made acceptable, moreover, by the work of Galileo,243 and Roberval and Torricelli were employing it in geometry at practically the same time as Descartes. Descartes' reasoning was as follows: If a polygon is rolled along a straight line, any vertex will describe a series of circular arcs, the centers of which are the points on the line which the vertices of the polygon touch: i. e., in rolling the polygon along the line, we rotate the polygon about each of these points in turn. The cycloid, now, is the curve generated by a point on a circle: i. e., by a vertex of a polygon with an infinite number of sides, as it is rolled along a line. The cycloid is therefore made up of an infinite number of circular arcs, and the tangent at any point, P is therefore perpendicular to the line joining P to the point Q in which the generating circle touches the base line. Inasmuch as ',... q, q', . . . , of the powers of h, k, . . . in the Taylor's series for u(x + h, y -f k, . . .) he defined as the "fonctions dérivées" of w. The differential calculus then consisted, i" Œuvres, XIV, 173; cf. IU, 443, and VII, 325-28. œ See Jourdain, "The Ideas of the 'Fonctions analytiques' in Lagrange's Early Work." »» See Œuvres, III, 439-76. Œuvres, III, 443.

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for Lagrange, in "finding directly and by simple and facile procedures the derived functions p, p', . . . , q, < ? ' , . . . of the function u"; and the integral calculus consisted, inversely, in "determining by means of these latter functions the function u."m Lagrange's method is based on the unwarranted supposition that every function can be so represented and handled. Moreover, the escape from the infinitely large and the infinitely small, as well as from the limit concept, is only illusory, inasmuch as these notions enter into the critical question of convergence which Lagrange did not adequately consider. Furthermore, his method lacks the operational suggestiveness and facility which the Leibnizian ideas and notation afforded. Logically, however, the Lagrangian definition had an advantage in that it sought, as had the work of Euler, to make fundamental the formalism of the theory of functions, rather than the preconceptions of geometry, mechanics, or philosophy. Lagrange has been criticized1™ for giving up, in favor of mathematical formalism, the "generative" concept which has frequently been felt to be the basis of the methods of fluxions and differentials. Such criticism is based on a failure to recognize that mathematics is most useful when unencumbered by psychological preconceptions. Euler had attempted misdirectedly to formalize the Leibnizian conceptions by making the differentials zeros. D'Alembert had made an effort to present a satisfactory idea of the notion of limit, but had failed to give the concept a clear and precise formalism which would make it logically unequivocal. Lagrange, therefore, sought still another mode of presentation, based on the function concept which Euler had emphasized and popularized. Incidentally, in so doing he focused attention for almost the first time upon the quantity which is now the central conception in the calculus—that of the derived function, or the derivative, or the differential coefficient (a terminology reminiscent of Lagrange's method, which was, however, introduced later by Lacroix).137 Lagrange, in this connection gave not only the name from which the word derivative was adopted, but also the notation f x , modifications of which are still conveniently used. m Ibid. u ' Cohen, Das Princif der Infiniiesimalmeihode und seine Gesckichte, p. 100. 137

See Biblioiheca Maihematica (3), I (1900), 517 for notes on the origin of these designations.

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Newton had emphasized the method of prime and ultimate ratios of increments or of fluxions. Although this ratio can be interpreted as a single number or quantity, which we now call the derivative, Newton seems to have had in mind the idea of this quantity as the ratio of increments or of fluxions, or of quantities proportional to these. Moreover, although the fluxions themselves can only be defined rigorously as derivatives, Newton does not appear to have realized this. In the differential calculus likewise it is clear that although Leibniz realized the significance of the ratio of two infinitesimals, he never seems to have thought of this ratio as a single number, but rather as a quotient of inassignables, or of assignables proportional to these. With D'Alembert's insistence that the differential calculus was to be rigorously interpreted only in terms of limits, one comes close to the conception of a derivative, but even here there is the lack of the notion of a single function, or number, obtained as the limit of a single infinite sequence. Like Newton and Leibniz, D'Alembert appears to have had in mind not a function, but two sides of an equation, the limits of which are equal. Kästner likewise displayed the predominance of the idea of a ratio. Although he recognized the fundamental significance of the limit concept, he nevertheless defined the differential quotient literally as a ratio of differentials.138 In Lagrange's method the term derivative can for the first time be applied with strict propriety, for his "fonction dérivée" is merely a single coefficient of a term in an infinite series and is completely divested of any idea of ratio, or limiting equality. It is a single quantity or function, and although Lagrange's definition was not that which was to be accepted in the end, the notion of a derived function may well have aided in making the general acceptance of the current definition possible. Although Lagrange's method appeared in the Miscellanea Taurinensia for 1772, it received little recognition—perhaps as a result of the novelty of the ideas and the notation involved—and the search for a satisfactory basis for the calculus continued. To encourage efforts in this direction the Berlin Academy, of which Lagrange at the time was president, in 1784 offered a prize for the best exposition of a clear 138

Anfangsgründe, p. 4.

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and precise theory of the mathematical infinity. The prize-winning essay was that of Simon L'Huilier: Exposition élémentaire des principes des calculs supérieurs. This was published in 1787 and appeared again (in Latin) in 1795. In this work L'Huilier proposed to show that "the method of the ancients, known under the name of Method of Exhaustion, conveniently extended, suffices to establish with certainty the principles of the new calculus."139 In conformity with this purpose, he modified the method of exhaustion to interpret it in terms of limits. Making the limit concept basic in his exposition, L'Huilier agreed with D'Alembert that "in the differential calculus it was not necessary to pronounce the name differential quantity." 140 As in modern textbooks, L'Huilier made the differential ratio or quotient fundamental, defining this as the limit of the ratio of the increment in the function to that in the independent variable. L'Huilier regarded this form of presentation of the calculus as a development of that of Newton and other English authors, but his exposition indicated an advance over their work. He focused attention upon a single number (the derivative) as the limit of a single variable (the ratio of the increments), rather than upon an ultimate ratio of two evanescent quantities, or of two fluxions, or of any two quantities which have the same ratio as these. Although he retained dy the name "differential quotient" and the symbol — to represent this dx quantity, he insisted that the latter was nothing but a symbol which was to be interpreted as a single number.141 In dealing with differential d2y was quotients of higher orders he again warned that the symbol — dx2 not to be broken up as a quotient. This is in striking contrast with the work of Newton and Leibniz, in which fluxions and differentials of any order were regarded as having a significance independent of the ratio or equation in which they entered. The differential quotient qf L'Huilier was a single number or function, equivalent to Lagrange's derived function, and represented essentially the present conception of the derivative. Although L'Huilier's definition of the differential quotient is in Exposition élémentaire, p. 6. Ibid., p. 141.

lw

»» Ibid., p. 32.

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most respects that to be found in present day elementary textbooks on the calculus, he does not seem to have been aware of the possible difficulties involved in the limit concept. He had avoided the mysticism of the infinitesimal, the vagueness of the ultimate ratio, and the inanity of the symbol jj; but he failed to appreciate that the subtlety of the limit concept was to make an extremely careful definition essential. L'Huilier was dealing only with very simple functions, so that he was unaware of the inadequacies of his presentation. His variable was always less than or greater than his limit. "Given a variable quantity always smaller or greater than a proposed constant quantity; but which can differ from the latter by less than any proposed quantity however small; this constant quantity is called the limit in greatness or in smallness of the variable quantity." 142 The variable could not oscillatc, as it may under our more general view. More seriously, L'Huilier fell into an error which is suggested by the vague idea of uniformity expressed in the law of continuity of Leibniz. He said that "if a variable quantity at all stages enjoys a certain property, its limit will enjoy this same property." 143 That this notion persisted also in the nineteenth century is apparent from the statement of William Whewell: "The axiom . . . that what is true u p to the limit is true at the limit, is involved in the very conception of a limit." 143 " The falsity of this doctrine is immediately apparent from the fact that irrational numbers may easily be defined as the limits of sequences of rational numbers, or from the observation that the properties of a polygon inscribed in a circle are not those of the limiting figure—the circle. The mistake of L'Huilier in this connection was probably the result of his failure to see that the limit concept was to be identified with the nature of infinite converging sequences and with the question of the nature of real numbers and the continuum. The inadequacy of his conception of number may be further inferred from the fact that he felt it necessary to distinguish between limiting values and limiting ratios. ia

Ibid., p. 7.

10

Exposition ilbneniaire, p. 167. Cajori ("Grafting of the Theory of Limits on the Calculus of Leibniz"), apparently unaware of this statement, has said that this principle was used but not stated in the eighteenth century. 1C

" History of Scientific Ideas, I., 152.

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Although L'Huilier correctly sought the basis of the calculus in the limit concept, his exposition of this was an oversimplification of what was later realized to be a very difficult question. Following D'Alembert in regarding the infinite from the point of view of magnitude rather than of aggregation, he denied the existence of an actually infinite quantity because he felt that its acceptance would lead to such "contradictions" as °° + « = 0 0 — n. He consequently maintained that he had shown the calculus to be independent of all idea of the infinite, whether large or small. In this he failed to realize that the whole theory of limits is based, in the last analysis, upon that of infinite aggregates. This fact was not clearly recognized until the following century. L'Huilier's monograph was not widely read, nor were his views generally accepted at the time. The indecision as to the true basis of the calculus remained as acute as before. As a result, there appeared in 1797 what was perhaps the most famous attempt to clear up the difficulties in the situation: the Réflexions sur la métaphysique du calcul infinitésimal, by L. N. M. Carnot, the remarkable soldier, administrator, and mathematician to whom the French Assembly gave the title "L'Organizateur de la Victoire." Carnot's work enjoyed a truly remarkable popularity, appearing in numerous editions and several languages from that time until quite recently. 1 " In view of the lack of clarity and uniformity in the then-current expositions of the calculus, Carnot wished to make the theory rigidly precise. Considering the many conflicting interpretations of the subject, he sought to know "in what the veritable spirit of the infinitesimal analysis consisted."146 In his selection of the unifying principle, however, he made a most deplorable choice. He concluded that "the true metaphysical principles of the Infinitesimal Analysis . . . are nevertheless . . . the principles of the compensation of errors,"146 as Berk144 The first edition of Paris, 1797, was followed by an enlarged second edition at the same place in 1813. Unless otherwise stated, references in this work are to this second edition. The first edition was translated into English by W. Dickson in the Philosophical Magazine, VIII (1800), 222^10; 335-52; I X (1801), 39-56. The second edition appeared in an English translation by W. R. Browell at Oxford, in 1832, as Reflexions on the Metaphysical Principles of the Infinitesimal Analysis. Other French editions appeared in Paris in 1839, 1860, 1881, and 1921 (2 vols.) A Portuguese translation appeared at Lisbon in 1798, a German one at Frankfurt a. M. in 1800, and an Italian one at Pavia in 1803. 144 Réflexions sur la métaphysique du calcul infinitésimal, p. 1. Browell trans., p. 44.

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eley and Lagrange had suggested. In his expansion of this view, he reverted substantially to ideas which Leibniz had expressed. He held that to be certain that two designated quantities are rigorously equal, it is sufficient to prove that their difference cannot be a "quantité designée."147 Paraphrasing Leibniz, Carnot said further: that for any quantity one may substitute another which differs from it by an infinitesimal;148 that the method of infinitesimals is nothing more than that of exhaustion reduced to an algorithm;149 that "quantités inappréciables" are merely auxiliaries which are introduced, like imaginary numbers, only to facilitate the computation, and which are eliminated in reaching the final result.160 Carnot even echoed the favorite explanation of Leibniz in terms of the law of continuity. He held that one can envisage the infinitesimal analysis under two points of view, according as the infinitesimals are taken to be "quantités effectives" or are regarded as "quantités absolument nulles." (Leibniz, however, had not admitted them as absolutely, but only relatively, zero.) In the first case, he felt that the calculus was to be explained upon the basis of a compensation of errors: "imperfect equations" were to be made "perfectly exact" by the simple expedient of eliminating the quantities whose presence occasioned the errors;161 in the latter case he considered the calculus an "art" of comparing vanishing quantities with each other in order to discover from these comparisons the relationships between the proposed quantities.162 T o the objection that these vanishing quantities either are or are not zero, Carnot responded that "what are called infinitely small quantities are not simply any null quantities at all, but rather null quantities assigned by a law of continuity which determines the relationship."163 This explanation is strikingly like that given by Leibniz about a century earlier. Along with his establishment of the "true metaphysics" of the infinitesimal calculus, Carnot proceeded to show that the diverse views of the subject were essentially reducible to this same basis. 147 Réflexions sur la métaphysique du calcul infinitésimal, p. 31. "» Ibid., p. 35. "» Ibid., p. 39. 1.0 Ibid., pp. 38-39. "Les mathématiques ne sont-elles pas remplies de pareilles énigmes?" said Carnot, in this connection. 1.1 Cf. Dickson's translation, p. 336. Réflexions sur la métaphysique du calcul infinitésimal, p. 185. "" Ibid., p. 190.

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In demonstrating this, he pointed out that the method of exhaustion made use of analogous systems of known auxiliary quantities. Newton's method of prime and ultimate ratios was similar, except that in it the lemmas freed the work from the need for the argument by a double reductio ad absurdum. Carnot felt that the methods of Cavalieri and Roberval also were admittedly corollaries of the method of exhaustion.154 Descartes' method of undetermined coefficients, he held, touched the infinitesimal analysis closely, the latter being but " a felicitous application" of the former!166 The method of limits he recognized to be not different from that of first and last ratios, so that it was likewise a simplification of the method of exhaustion. 166 Furthermore, these methods all lead to the results of infinitesimal analysis, but by a difficult and circuitous route.167 Lagrange's method he likewise saw linked to the infinitesimal calculus, in that it neglects the other terms of the infinite series.168 The divers points of view he therefore felt, somewhat as had L'Huilier, to be merely simplifications of the method of exhaustion which were effected by reducing this to a convenient algorithm. Inasmuch as the infinitesimal method combined the facility of the procedures of approximate calculation with the exactitude of the results of ordinary analysis, he saw no point in attempting, under the guise of greater rigor, to substitute for it any less natural method.159 Although Carnot's work was widely read, it can hardly be said to have led to a clearer understanding of the difficulties inherent in the new analysis. Although he realized that differentials were to be defined somewhat as variables, anticipating, to a certain extent, the view of Cauchy, he was unable to give suitable definitions to these because, like Leibniz, he thought in terms of equations rather than of the function concept which led Cauchy to make the derivative fundamental. Furthermore, Carnot, one of a school of mathematicians who emphasized the relationship of mathematics to scientific practice, 160 appears, in spite of the title of his work, to have been more concerned about the facility of application of the rules of procedure than about the logical reasoning involved. In this respect his work resembles ut ™Ibid., pp. 139-40. Ibid., pp. 150-51. Ibid., p. VI. 117 Ibid., p. 192. ' « I b i d . , pp. 194-97. "»Ibid., pp. 215-16. 1,0 Merz, A History of European Thought in the Nineteenth Century, II, 100-1.

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that of Leibniz, whose explanations of differentials he so largely paraphrased, and whose method he defended with almost polemic warmth. Mathematicians on the Continent had been in essential agreement that pragmatically the differential calculus offered the best method of procedure; but it was precisely on the logic of the matter that they were at variance. On the latter point Carnot was of little assistance, for after pointing out what most men had long realized—that all of the methods of the new analysis were essentially related—he proceeded to make basic the infinitesimal system, the one which was logically, perhaps, the weakest of all. In so doing he pointed toward a view diametrically opposed to that which D'Alembert161 and L'Huilier had indicated and along which the rigorous development was ultimately to proceed, following the work of Cauchy.162 The year 1797, in which the first edition of Carnot's work was published, saw the appearance also of the famous work of Lagrange, Théorie des fonctions analytiques, contenant les principes du calcul différentiel, dégagés de toute considération d'infiniment petits ou d'évanouissans, de limites ou de fluxions, et réduits à l'analyse algébrique des quantités finies. This book developed with care and completeness the characteristic definition and method in terms of "fonctions dérivées," based upon Taylor's series, which Lagrange had proposed in 1772. In it the author gave not only an attempted proof of the incorrect theorem that every continuous function may be so expanded, but also the determination of the "fonctions derivées" (or derivatives) of the elementary functions, and numerous applications to geometry and mechanics. Carried along by the authority of Lagrange's great reputation, the method now enjoyed a short-lived period of comparative success. As had been the case with Maclaurin's treatise on fluxions, mathematicians praised the new method highly,163 although they seldom used it. They explained that inasmuch as the notations were 1.1 Mansion (Esquisse de l'histoire du calcul infinitésimal, p. 290, n.) has misdirectedly characterized Carnot's ideas as a development of those of D'Alembert. 1.2 The assertion (See Smith, "Lazare Nicholas Marguérite Carnot," p. 189) that Carnot "paved the way for Cauchy's notable memoir" is justified only in the most general sense. 1M Cf. the review in Monthly Review, London, N. S., XXVIII (1799), appendix, pp. 48199; also Valperga-Caluso, "Sul paragone del calcolo delle funzioni derivate coi metodi anteriori"; also the set of monographs by Froberg, and others, called De analytica calculi differentialis et integralis theoria, inventa a cel. La Grange.

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less convenient and the calculations involved more embarrassing than those found in the methods of the ordinary differential and integral calculus, it was sufficient that one were assured through Lagrange's method of the legitimacy of other more expeditious methods. This seems to have been the view of Lagrange himself, for side by side with his method of derived functions he continued throughout his life to employ the notation of differentials. Most of the objections to the method of Lagrange were based upon the inconveniences of the notation and the operations, but before long doubts began to arise as to the correctness of the principle that all continuous functions could be expanded in Taylor's series. It was pointed out164 that such an expansion was possible only for the more simple functions, and that consequently the method was of limited applicability. Furthermore, Sniadecki, a Pole, correctly explained that the method was fundamentally identical with the limit method. 165 However, the views of convergence, continuity, and function held at the time were not sufficiently definite to permit a deeper clarification of these ideas. An interesting but somewhat misdirected tirade against Lagrange's point of view was delivered by another Polish mathematician, Hoëné Wronski. An eager devotee of the differential method of Leibniz and of the transcendental philosophy of Kant, he protested with some asperity against the ban on the infinite in analysis which Lagrange had wished to impose. He criticized Lagrange not so much for the absence of logical rigor in his free manipulation of infinite series— although he did pertinently ask where Lagrange got the series f(x + i) = A + Bi + Ci2 + Di3 + . . . with which he opened his proof of Taylor's series—as for his lack of a sufficiently broad view. Wronski believed that modern mathematics is to be based on the "supreme algorithmic law" Fx = ^40Oo + Aiii + A-£h + A3Çl3 + • • • , where the quantities £2o, fii, O2, ÎÎ3, • • • are any functions of the variable x. Being the supreme law of mathematics, the irrecusable truth of this law he held to be not mathematically derived, but given by transcendental philosophy.16® 1M See Dickstein, "Zur Geschichte der Prinzipien der Infinitesimalrechnung"; cf. also Lacroix, Traité du calcul, 2d éd., I l l , 629-30. 1,4 Dickstein, "Zur Geschichte der Prinzipien der Infinitesimalrechnung." Réfutation de la théorie des fonctions analytiques, Vol. IV of Wronski's Œuvres.

262

The Period of Indecision

Wronski was, of course, correct in saying that the method of derived functions was too restricted, in that it was limited only to certain functions which could be so expanded. However, his general views on the calculus were far from those accepted at the present time. Whereas Lagrange had attempted to give a formal logical justification of the subject, Wronski asserted that the differential calculus constituted a primitive algorithm governing the generation of quantities, rather than the laws of quantities already formed. Its propositions he held to be expressions of an absolute truth, and the deduction of its principles he consequently regarded as beyond the sphere of mathematics. The explication of the calculus by the methods of limits, of ultimate ratios, of vanishing quantities, of the theory of functions, he felt constituted but an indirect approach which proceeded from a false view of the new analysis. In seeking to examine the principles of the subject by purely mathematical means, Wronski believed geometers were simply wasting time and effort. He called upon them to give up "that servile imitation of ancient geometers in avoiding the infinite and that which depends upon it."167 In his own work Wronski made a more uncritical use of the infinite than had even Wallis and Fontenelle. He said, for example, that "the absolute meaning of the number r was given by the expression . . . 4 0° f (1 + V - 1)5 - (l _ V - 1)41 ."«a perhaps because of the V " , novelty of his notation, as well as of this bizarre use of the symbol 00 , Wronski did not exert a strong influence upon the development of the calculus. The mathematics of the time was about to accept what he opposed: the rigorous logical establishment of the calculus upon the limit concept. Nevertheless, the work of Wronski represents an extreme example of a view which we shall find recurring throughout the nineteenth century. In regarding the calculus as a means of explaining the growth of magnitudes, followers of this school of thought were to attempt to retain the concept of the infinitely small, not as an extensive quantity but as an intensive magnitude. Mathematics has excluded the fixed infinitely small because it has failed to establish the notion logically; but transcendental philosophy has sought to preserve primitive intuition in this respect by interpreting it as Ibid., pp. 39-40.

1M

Ibid., p. 69.

The Period of Indecision

263

having an a priori metaphysical reality associated with the generation of magnitude. Lagrange's Théorie des fonctions was only one, but by far the most important, of many attempts made about this time to furnish the calculus with a basis which would logically modify or supplant those given in terms of limits and infinitesimals. Condorcet, Arbogast, Servois, and others put forth methods similar to that of Lagrange. 16 ' Condorcet, as early as 1786, had begun a work on the calculus, based on series and finite differences, but this was interrupted by the Revolution and did not appear.170 In 1789 L. F. A. Arbogast presented to the Académie des Sciences a "true theory of the differential calculus" along the lines of Lagrange and Condorcet. This work, published in 1800 under the title Du calcul des dérivations, sought to establish the principles of the calculus independently of limits and the infinitely small, and with the simplicity and certitude found in ordinary algebra.171 Arbogast assumed, as had Lagrange, that the function F(a + x) could be expanded in a series of powers of x, and then showed that the coefficients in this could be identified with the more familiar differential quotient.172 These new methods resemble the attempts which had been made in England, following the Berkeley-Jurin-Robins dispute, to give arithmetical procedures for the calculus. They serve to indicate, as did the British controversy, a lack of satisfaction, at the time, with the methods of limits, fluxions, and infinitesimals. The full title of Lagrange's Théorie des fonctions indicates his discontent with these methods. F. J . Servois, in his Essai sur un nouveau mode d'exposition des principes du calcul différentiel, which appeared in 1814,173 expressed himself more strongly. He called the method of limits a "gothique hypothèse"174 and that of differentials a "strabisme infinitésimal."175 1M For references to this work, see Dickstein, op. cit.; Moritz Cantor, Vorltsungen, IV, Section X X V I ; and Lacroix, Traité du calcul différentiel et du calcul intégral, I, 237-48, and preface. 170 See Lacroix, Traité du calcul, I, xxii-xxvi. 171 See Zimmermann, Arbogast als Mathematiker uni Historiker der Maihematik, pp. 44-45; cf. also Arbogast, Du calcul des dérivations, Preface; and Lacroix, Traité du calcul, Preface. m Du calcul des dérivations, pp. xii-xiv; cf. also p. 2. 171 This appeared in the Annales de Mathématiques, V (1814-15), pp. 93-141, and was also published separately in Nîmes, 1814. References given in this work are to the Nlmes publication. m Ibid., p. 65. Essai, p. 56.

264

The Period of Indecision

The introduction of the idea of infinity he objected to as useless; and, with reference to the views which Wronski had expressed on the calculus in his Réfutation of Lagrange, he asserted that he "had well foreseen, on reading Kant, that geometers would sooner or later be the object of the cavils of his sect." 17 ' Servois tried, in his turn, to establish the calculus on a combination of finite differences and infinite series, in which the idea of limits was, as in the case of the method of Lagrange, ostensibly eliminated by a disregard of the essential question of convergence. Further attempts along this line continued for sometime: in 1821 in Eine neue Methode jiir den Infinitesimalkalkul of Georg von Buquoy ; in 1849 in the Essai sur la métaphysique du calcul intégral of C. A. Agardh; and as late as 1873 in an article, "Exposition nouvelle des principes de calcul différentiel," by J. B. Brasseur. 177 These invariably depended upon expansions in series to avoid "any metaphysical notions," and in this respect they resembled closely the earlier efforts of Lagrange, Condorcet, Arbogast, and Servois. However, at almost precisely the period of the work of Servois a tendency toward rigor in series was setting in. This made clear the basic weakness of all such attempts to avoid the method of limits through an uncritical formal manipulation of infinite series. The final formulation of the calculus was not destined to be in terms of one of these new methods which were developed, but was to be based on one of the very notions they sought to avoid—that of a limit. In the year 1797, in which Carnot attempted a concordance of all the systems of the calculus and in which Lagrange tried seriously to establish his method, there was published the first volume of perhaps the most famous and ambitious textbook on the subject which had appeared up to that time—Traité du calcul différentiel et du calcul intégral of Lacroix. Lacroix declared in the preface of this work that it had been the new method of Lagrange which had inspired him to compose a treatise on the calculus which should have as a basis the luminous ideas which this method had substituted for the infinitely small. His text, however, indicates well the indecision of the period, for in spite of the professed aim of the work, the foundation of the subject remained somewhat in doubt. Lacroix interpreted the method m

Ibid., p. 68.

177

See Bibliography below, for full citations of these works.

The Period of Indecision

265

of series of Lagrange in terms of the limits of D'Alembert and L'Huilier. In this respect, however, he obscured the significance of this relationship by speaking of the limit of divergent series, and by following Euler in the study of such infinite series as 1 — 1 + 2 — 6 + 24 — 120 + . . ,178 Furthermore, he did not share with Lagrange the suspicion that the method of Leibniz was based on a false idea of the infinitely small. Lacroix admittedly made use of infinitesimals. 17 ' Although he accepted the metaphysics of Lagrange, he adopted the differential notation of Leibniz.180 This fact at times confused his thought and led him, as it had Euler, to regard the differential coefficient (as Lacroix called it) as a quotient of zeros.181 The lack on the part of Lacroix of critical distinction in dealing with the methods of Newton, Leibniz, and Lagrange gave to his work an appearance somewhat resembling the attempt of Carnot to demonstrate the congruity of the numerous representations of the calculus. The mathematician and astronomer Laplace praised this attitude on the part of Lacroix, saying that such a rapprochement of methods served as a mutual clarification, the true metaphysics probably being found in what they had in common.182 It appears, however, that at that time at least it was deplorable, for it led to a confusion of thought on the subject just when logical precision was most needed. However, in his more popular work of 1802—the Traité élémentaire, an abridgment of his larger treatise—Lacroix omitted the method of Lagrange and made the explanation in terms of limits basic, although again with a lack of rigor which made interpretations in terms of infinitesimals possible. The success of this work, which went through many editions (the ninth French edition appearing in 1881) and was translated into several languages, led to other texts of the same type; and it was largely through these that the method of limits became familiar, if not rigorous. It was through such texts that the Leibnizian notation and the doctrine of limits supplanted in England the method of fluxions and interpretations which had already become hopelessly confused with the infinitely small. The year 1816, in which Lacroix's shorter work was translated into m

Traité du calcul, III, 389. 1,1 "» Ibid., I, 242. ™ Ibid., p. 243. Ibid., I, 344. UB Letter to Lacroix, published in Le Nouveau Traité du calcul dijférenlid, 2d éd., preface, p. X I X .

266

The Period of Indecision

English, "marks an important period of transition," 183 because it witnessed the triumph in England of the methods used on the Continent. This particular point in the history of mathematics marks a new epoch for a far more significant reason, for in the very next year there appeared a work by Bolzano—Rein analytischer Beweis des Lehrsatzes dass zwischen je zwei Werthen, die ein entgegengesetztes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege— which indicated the rise of the period of mathematical rigor in all branches of the subject.144 In the calculus the new attitude resulted in the logical establishment of the higher analysis upon the limit concept, and thus brought to an end the period of indecision which had begun with the inventions of the method of fluxions and of the differential calculus. 10 1,4

Cajori,

A History of the Conceptions of Limits and Fluxions, See Pierpont, "Mathematical Rigor," pp. 32-34.

pp. 270-71.

VII. The Rigorous Formulation

T

HE OBJECTIONS raised in the eighteenth century to the methods of fluxions, of prime and ultimate ratios, of limits, of differentials, and of derived functions were in large measure unanswered in terms of the conceptions of the time. The arguments were in the last analysis equivalent to those which Zeno had raised well over two thousand years previously and were based on questions of infinity and continuity. The proponents of all methods except that of differentials, however, protested that they had no need to invoke the notion of the infinite and disregarded entirely that of continuity. The advocates of the differential calculus, in turn, although attempting to justify its procedures in terms of these concepts, were quite unable to furnish logically consistent explanations of them. By most mathematicians these were considered to be metaphysical ideas and so to lie beyond the realm of mathematical definition. It is interesting, in looking back, to see that the methods most adverse to the introduction into mathematics of the notions of infinity and of continuity were precisely those which made this introduction possible. The method of limits, which at that time appeared to lead to neither infinity nor the law of continuity, was to furnish the logical basis for these; and the method of Lagrange, which was developed in order to avoid these difficulties, was to bring up questions which pointed the way toward their solution. We have seen that in the early nineteenth century critics of Lagrange began to question the validity of his principle that a continuous function can always be expressed, by means of Taylor's theorem, as an infinite series. They began to ask what was meant by a function in general and by a continuous function in particular, and to criticize the almost indiscriminate use of infinite series. Lagrange had made a start in the direction of greater care in the use of series when he pointed out that one must consider the remainder in every case. This warning serves to indicate, perhaps, why he thought he had avoided the use of the infinite and the infinitely small. Like Archimedes, he evidently did not consider the series as continued to infinity, but only to a point at which the remainder

268

The Rigorous Formulation

was sufficiently small. In the nineteenth century, however, the concept of infinity was to become basic in the calculus through the use of infinite series and infinite aggregates. One of the pioneers in the matter of greater rigor in the fundamental conceptions of the calculus—in its arithmetization and in the careful study of the infinite—was the Bohemian priest, philosopher, and mathematician, Bernhard BoLzano.1 In 1799 Gauss had given a proof of the fundamental theorem of algebra—that every rational, integral algebraic equation has a root—using considerations from geometry. Bolzano, however, wished a proof involving only considerations derived from arithmetic, algebra, and analysis. As Lagrange had felt that the introduction of time and motion into mathematics was unnecessary, so Bolzano sought to avoid in his proofs any considerations derived from spatial intuition. 2 This attitude made necessary, in the first place, a satisfactory definition of continuity. The calculus may, indeed, be thought of as having arisen from the Pythagorean recognition of the difficulty involved in attempting to substitute numerical considerations for supposedly continuous geometrical magnitudes. Newton had avoided such embarrassment by appealing to the intuition of continuous motion, and Leibniz had evaded the question by his appeal to the postulate of continuity. Bolzano, however, gave a definition of continuous function which, for the first time, indicated clearly that the basis of the idea of continuity was to be found in the limit concept. He defined a function j{x) as continuous in an interval if for any value of x in this interval the difference j{x + A*) — f(x) becomes and remains less than any given quantity for A x sufficiently small, whether positive or negative. 3 This definition is not essentially different from that given a little later by Cauchy and is fundamental in the calculus at the present time. In presenting the elements of the calculus, Bolzano realized clearly that the subject was to be explained in terms of limits of ratios of 1 See Stolz, "B. Bolzano's Bedeutung in der Geschichte der Infinitesimalrechnung." Although Bolzano was born and died at Prague, Quido Vetter ("The Development of Mathematics in Bohemia," p. 54) has called him a Milanese, inasmuch as his father was a native of northern Italy. * Rein analytischer Beweis, pp. 9-10. ' See Bolzano's Schriften, I, 14; cf. also Rein analytischer Beweis, pp. 11-12.

The Rigorous Formulation

269

finite differences. He defined the derivative of F(x) for any value of F(x + Ax) - F(x) x as the quantity F'{x) which the ratio approaches Ax indefinitely closely, or as closely as we please, as Ax approaches zero, whether Ax is positive or negative. 4 This definition is in essence the same as that of L'Huilier, but Bolzano went further in explaining the nature of the limit concept. Lagrange and other mathematicians had felt that the limit notion was bound up with a quotient of evandy escent quantities or of zeros. Euler had explained — as a quotient of dx zeros, and in this respect Lacroix tended to follow him. Bolzano, however, emphasized the fact that this was not to be interpreted as a ratio of dy to dx or as the quotient of zero divided by zero, but was rather one symbol for a single function. 4 He held that a function has no determined value at a point if it reduces to jj. However, it may have a limiting value as this point is approached, and he correctly indicated that, by adopting the limiting value as the meaning of the function may be made continuous at this point. 6 Ever since the invention of the calculus it had been felt that, inasmuch as the subject was bound up with motion and the growth of magnitudes, the continuity of a function was sufficient to assure the existence of a derivative. In 1834, however, Bolzano gave an example of a nondifferentiable continuous function. This is based upon a fundamental operation which may be described as follows: Let PQ be a line segment inclined to the horizontal. Let it be halved by the point M, and subdivide the segments PM and MQ into four equal parts, the points of division being Pu P 2 , P 3 , and Qt, Q2, Qi. Let Pi be the reflection of P3 in the horizontal through M, and let Qi be the reflection of ft in the horizontal through Q. Form the broken line PP3 'MQ3 'Q. Now apply to each of the four segments of this broken line the fundamental operation as described above, obtaining in this manner a broken line of 42 segments. Continuing 4 Schriften, I, 80-81; cf. Paradoxien des Unendiichen, p. 66. • Paradoxien des Unendiichen, p. 68.

* Schriften, pp. 25-26.

270

The Rigorous Formulation

this process indefinitely, t h e broken-lined figure will converge toward a curve representing a continuous function nowhere differentiable. 7 T h i s illustration given by Bolzano might have served in mathematics as the analogue of the experimentum crucis in science, showing t h a t continuous functions need not, in spite of the suggestions of geometrical and physical intuition, possess derivatives. However, because the work of Bolzano did not become known at t h e time, such a role was reserved for t h e famous example of such a function given by Weierstrass about a t h i r d of a century later. 8 Lagrange had held t h a t his method of series avoided t h e necessity of considering the infinitesimals or limits, b u t Bolzano pointed out t h a t in t h e case of infinite series it is necessary to consider questions of convergence. These are analogous to limit considerations, as is obvious from the s t a t e m e n t of Bolzano t h a t if the sequence F\{x), F2(X), F3(X), . . . , Fn{x), . . . , . . . is such t h a t t h e difference between Fn(x) and FN + r(x) becomes and remains less t h a n any given quantity as n increases indefinitely, then there is one a n d only one value to which the sequence approaches as closely as one pleases.® This fundamental proposition of Bolzano will be seen later to have also a significance with respect to the general definition of real number and the arithmetical continuum. Bolzano felt, in spite of the paradoxes presented by t h e notions of space and time, t h a t a n y continuum was to be t h o u g h t of as ultimately composed of points. 1 0 His view in this respect resembles t h a t of Galileo, to whom he referred in this connection. 11 Although he denied the existence of infinitely large a n d infinitely small magnitudes, he maintained, with Galileo, the possibility of an actual infinity with respect to aggregation. H e remarked, with respect to such assemblages, the paradox which Galileo had pointed o u t : t h a t the p a r t could in this case be p u t into one-to-one correspondence with the whole. T h e numbers between 0 and 5, for example, could be paired with those between 0 a n d 12.12 Bolzano's views on the infinite are substantially those which mathematicians have adopted since the 7

See Kowalewski, "Uber Bolzanos nichtdifferenzierbare stetige Function." Waismann (Einführung in das mathematische Denken, p. 122) mistakenly represents Weierstrass as the first to give such an example. 10 Paradoxien des Unendlichen, pp. 75-76. * Rein analytischer Beweis, p. 35. u u Ibid., pp. 89-92. Ibid., pp. 28-29. 8

The Rigorous Formulation

271

time of Cantor, except that what Bolzano had considered as different powers of infinity have been discovered to be of the same power. Bolzano, however, sought to prove the existence of the infinite upon theological grounds, whereas later in the century the property which he and Galileo had regarded simply as a paradox was made by Dedekind, in his clarification of the calculus, the basis of a definition of infinite assemblages. Although Bolzano's ideas indicate the direction in which the final formulation of the calculus lay and which much of the thought of the nineteenth century was to follow, they did not constitute the decisive influence in determining this. His work remained largely unnoticed until rediscovered by Hermann Hankel more than a half century later. 13 Fortunately, however, the mathematician A. L. Cauchy pursued similar ideas at about the same time and was successful in establishing these as basic in the calculus. Cauchy rivaled Euler in mathematical productivity, contributing some 800 books and articles on almost all branches of the subject.14 Among his greatest contributions are the rigorous methods which he introduced into the calculus in his three great treatises: the Cours d'analyse de l'École Polytechnique (1821), Résumé des leçons sur le calcul infinitésimal (1823), and Leçons sur le calcul différentiel (1829).16 Through these works Cauchy did more than anyone else to impress upon the subject the character which it bears at the present time. We have seen the notion of a limit develop gradually out of the Greek method of exhaustion, until it was expressed by Newton in the Principia. It was more definitely invoked by Robins, D'Alembert, and L'Huilier as the basic concept of the calculus, and as such was included by Lacroix in his textbooks. Throughout this long period, however, the limit concept lacked precision of formulation. This resulted from the fact that it was based on geometrical intuition. I t could hardly have been otherwise, inasmuch as during this time the ideas of arithmetic and algebra were largely established upon those of geometrical magnitude. The calculus was interpreted by its inventors as an instrument for dealing with relationships between u

See Dickstein, "Zur Geschichte der Principien der Infinitesimalrechnung," p. 77; also

Hankel's article "Grenze," in Ersch und Gruber's Allgemeine Encyklopadie. " See Valson, La Vie et les travaux du Baron Cauchy, reviewed by Boncompagni, p. 57. 16 The first is found in Œuvres (2), III; the last two in Œuvres (2), IV.

272

The Rigorous Formulation

quantities involved in geometrical problems, and as such was largely accepted by their successors. Euler and Lagrange, in a sense represented exceptions to this rule, for they wished to establish the calculus on the formalism of their analytic function concept. However, even they rejected the limit idea. Furthermore they unwittingly deferred to the preconceptions of geometrical intuition when they uncritically inferred t h a t their methods were applicable to all continuous curves. Although D'Alembert, L'Huilier, and Lacroix had prepared the ground for Cauchy by popularizing the limit idea in their works, this conception remained largely geometrical. In the work of Cauchy, however, the limit concept became, as it had in the thought of Bolzano, clearly and definitely arithmetical rather t h a n geometrical. Formerly, when illustrations of the notion were desired, the one most likely to be called to mind was t h a t of a circle defined as the limit of a polygon. Such an illustration immediately served to bring up questions as to the manner in which this is to be interpreted. Is it the approach to coincidence of the sides of the polygon with the points representing the circle? Does the polygon ever become the circle? Are the properties of the polygon and the circle the same? I t was questions such as these t h a t retarded the acceptance of the limit idea, for they were similar to those of Zeno in demanding some sort of visualization of the passage from the one to the other by which the properties of the first figure merge into those of the second. Quite recently the limit concept has been loosely interpreted in the assertion that whether one calls the circle the limit of a polygon as the sides are indefinitely decreased, or whether one looks upon it as a polygon with an infinite number of infinitesimal sides, is immaterial, inasmuch as in either case in the end "the specific difference" between the polygon and the circle is destroyed. 16 Such an appeal to geometrical intuition is quite irrelevant in the case of the limit concept. In giving his definition, in his Cours d'analyse, Cauchy divorced the idea from all reference to geometrical figures or magnitudes, saying: "When the successive values attributed to a variable approach indefinitely a fixed value so as to end by differing from it by as little as one wishes, this last is called the limit of all the others." 1 7 " Vivanti, II concetto d'infinitesimo, p. 39.

17

See Œuvres (2), III, 19; cf. also IV, 13.

The Rigorous Formulation

273

This is the most clear-cut definition of the concept which had been given up to that time, although later mathematicians were to voice objections to this also and to seek to make it still more formal and precise. Cauchy's definition appealed to the notions of number, variable, and function, rather than to intuitions of geometry and dynamics. Consequently, in illustrating the limit concept he said that an irrational number is the limit of the various rational fractions which furnish more and more approximate values of it." Upon the basis of this arithmetical definition of limit, Cauchy then proceeded to define that elusive term, infinitesimal. Ever since the time that Greek mathematical speculation had hit upon the infinitely small, this notion had been bound up with geometrical intuition of spatial properties and had been regarded as a more or less fixed minimum of extension. The concept had not thrived in arithmetic, largely because unity was considered the numerical minimum, the rational fractions having been treated as ratios of two numbers. However, the seventeenth century saw the rapid rise of algebraic methods in geometry, so that from the time of Fermât the infinitesimal had been the concern of both algebra and geometry. Newton had insisted that his method did not involve the consideration of minima sensibilia, but his interpretation of the procedure in terms of an ultimate ratio, rather than of a limiting number, made his evanescent magnitudes appear such. Liebniz had been less definite and consistent in denying the existence of actual infinitesimals, for he sometimes considered them as assignable, sometimes as inassignable, and occasionally as qualitatively zero. However, the development during the eighteenth century of the function concept, with its emphasis on the relations between variables, led Cauchy to make the infinitesimal nothing more than a variable. "One says that a variable quantity becomes infinitely small when its numerical value decreases indefinitely in such a way as to converge toward the limit zero." 19 An infinitesimal was consequently not different from other variables, except in the understanding that it is to take on values converging toward zero as a limit. In order to make the concept of the infinitesimal more useful and to take advantage of the operational facility afforded by the Leibnizian u

Œuvres (2), III, 19, 341.

« Œuvres (2), HI, 19; IV, 16.

274

The Rigorous Formulation

views, Cauchy added definitions of infinitesimals of higher order. Newton had restricted himself to infinitesimals, or evanescent quantities, of the first order, but Leibniz had attempted to define those of higher orders also. One of the second order, for example, he defined as being to one of first order as the latter is to a given finite constant, such as unity. Such a vague definition could not be consistently applied, but D'Alembert had sought to correct it by an interpretation in terms of limits. He had in a sense recognized that the infinitesimals were to be regarded as variables and that orders of infinitely small magnitudes were to be defined in terms of ratios of these; but his work, as we have seen, lacked the precision of statement necessary for its general acceptance. It thus remained for Cauchy to express the idea of D'Alembert in the precise symbolism of the limit concept. Cauchy defined an infinitesimal y = f(x) to be of order n with respect to an infinitesimal x if**™o(

I = 0 and e

I = =*= n+

y-+o\x"- J

y-*o\x

00

,

)

e having its classical significance—a positive constant, however small.20 Here again one recognizes in Cauchy's work the dominance of the ideas of variable, function, and limit. Newton, Leibniz, and D'Alembert had not distinguished clearly between independent and dependent infinitesimals, but Cauchy incorporated this in his definition when he spoke of the order of the infinitesimal, y, with respect to another, x. Thus the latter is the independent variable which may be given any sequence of values tending toward 0 as a limit, the corresponding sequence of values of y being determined from the functional relationship between y and x. Then the limits of the y sequences of values of the ratios ——• and x

v

are found, and the

x

order of the infinitesimal y thus determined. This is substantially the same as the definition commonly given in present-day textbooks— that y is said to be an infinitesimal of order n with respect to another infinitesimal x if

—m is a constant different from zero.

In a similar manner Cauchy made rigorously clear the views on orders of infinity which D'Alembert had expressed. Whereas Bolzano, » See Œuvres (2), IV, 281.

The Rigorous Formulation

275

thinking in terms of aggregation, had asserted the possibility of an actual infinity, Cauchy, emphasizing variability, denied the possibility of this because of the paradoxes to which such an assumption appeared to lead.21 He admitted only the potential infinite of Aristotle, and with D'Alembert interpreted the infinite as meaning simply indefinitely large—a variable, the successive values of which increase beyond any given number. 22 Orders of infinity he then defined in a manner exactly analogous to that given for infinitesimals. Having established the notions of limit, infinitesimal, and infinity, Cauchy was able to define the central concept of the calculus—that of the derivative. His formulation was precisely that given by Bolzano: Let the function be y = f(x); to the variable x give an increment A x = z; and form the ratio

= fS-h-Q Ax

The li m it of this i

ratio ("when it exists") as i approaches zero he represented by f'(x), and he called this the derivative of y with respect to xP This is, of course, the differential quotient of L'Huilier, clarified by the application of the function concept of Euler and Lagrange. It is made the central concept of the differential calculus, and the expression "differential" is then defined in terms of the derivative. The differential thus represents simply a convenient auxiliary notion permitting the application of the suggestive notation of Leibniz without the confusion between increments and differentials which this symbolism had engendered. Leibniz had considered differentials as the fundamental concepts, the differential quotient being defined in terms of these; but Cauchy reversed this relationship. Having defined the derivative in terms of limits, he then expressed the differential in terms of the derivative. If dx is a finite constant quantity h, then the differential dy of y = f ( x ) is defined as f'{x)dx. In other words, the differentials (dy) dy and dx are quantities so chosen that their ratio —

. coincides

{dx)

with that of the "dernière raison" or the limit y' = J'(x) of the ratio Ay ,24 This is practically the view of D'Alembert and L'Huilier, and — Ax 21

See Enriques, Historic Development of Logic, p. 135. Œuvres (2), HI, 19 ff.; cf. also IV, 16. " Œuvres (2), IV, 22; cf. also pp. 287-89. » Œuvres (2) IV, 27-28; 287-89. a

276

The Rigorous Formulation

even Leibniz had in a sense anticipated it when he said in 1684 t h a t dy was to dx as the ratio of the ordinate to the subtangent. T o make his work logical, however, Leibniz would have had to define the term " s u b t a n g e n t " in terms of limits, thus making the limiting ratio the derivative. Cauchy, however, gave to the derivative and the differential a formal precision which had been lacking in the definitions of his predecessors. He was therefore able to give satisfactory definitions of differentials of higher order also. The differential dy = J'(x)dx is, of course, a function of x and dx. Regarding dx as fixed, the function f'(x)dx will in turn have a derivative f"(x)dx and a differential d^y = Cauchy added t h a t because the f"(x)dx} In general Opera omnia, XI, 7. » Exposition élémentaire, p. 32. Cf. p. 144. » Œuvres, III, 443.

The Rigorous Formulation

279

derived." Bolzano similarly defined the integral as the inverse of the derivative.34 The result of this tendency in the calculus was that logically the definition of the integral during this time rested immediately upon that of the differential, and the latter became the basis of discussions on the validity of the operations and conceptions of the calculus. Developments during the early nineteenth century, however, introduced new points of view. These led to the reinstatement by Cauchy of the notion of the (definite) integral as a limit of a sum, and made necessary two independent definitions of the two fundamental concepts of the calculus, those of the derivative and the integral. Inasmuch as the derivative has been defined as

^ h we see from the definition of continuity given by Bolzano and Cauchy that the existence of this implied the continuity of the function at the value in question, although the converse is not true. The existence of an integral in the eighteenth-century sense, that is, as an antiderivative, is therefore bound up with the question of continuity. However, even discontinuous curves apparently have an area, and so discontinuous functions may allow of an integral in the Leibnizian sense. Cauchy therefore restored the character of the definite integral as a sum. For a function y = fix), continuous in the interval from x0 to X, he formed the characteristic sum of the products Sn = (*, - *b)/(*0) + (*» - *i)/(*i) + . . . + (X - * „ _ , ) / ( ! . _ ! ) . If the absolute values of the differences xi + l — decrease indefinitely, the value of Sn will "finally attain a certain limit" S which will depend uniquely on the form of the function fix) and on the limiting values x0 and X . . . "This limit is called a definite integral." 36 Cauchy cautioned that the symbol of integration / employed to designate this limit was not to be interpreted as a sum, but rather as a limit of a sum of this type.36 Cauchy then brought out the fact that although the two operations are defined independently of each other, integration in this sense is the inverse of the process of differ- Traili du calcul, " Œuvres (2), IV, 664 ff.; "The Theory " Œuvres (2), IV,

II, 1-2. " Bolzano's Schriften, pp. 83-84. 125; cf. also Jourdain, "The Origin of Cauchy's Conceptions," pp. of Functions with Cauchy and Gauss," p. 193. 126.

280

T h e Rigorous Formulation

entiation. He showed that if f(x) is a continuous function, the function defined as the definite integral F(x) = fl0j(x)dx has as its derivative the function f(x).17 This was perhaps the first rigorous demonstration of the proposition known as the fundamental theorem of the calculus." Cauchy's definition of the definite integral allows of extension, with slight modifications, to functions which have discontinuities within the interval of definition. If, for example, the function f(x) is discontinuous at the point Xt> within the interval x0 to X, the definite integral from x0 to X is defined as the limit, if it exists, of the sum in

~ *J(x)dx

+

fxo + tf(x)dx,

a s e b e c o m e s i n d e f i n i t e l y small. 3 9

Discontinuous functions have come to play a significant rôle in mathematics and science, and the view of the integral as a sum has been that upon which the theory of integration has largely developed since the time of Cauchy. From this view, for example, the integral of Lebesgue has developed.40 The manipulation of infinite series, such as those entering in the definition of the definite integral, had gone on for well over a century before the time of Cauchy, but the need for considering the convergence of these had not been strongly felt until about the opening of the nineteenth century. The term "convergent series" seems first to have been used, although in a somewhat restricted sense, a century and a half earlier by James Gregory; 41 but the general lack of rigor in the work of the eighteenth century was uncongenial to the precision of thought necessary to develop this idea. Varignon, at the beginning of the century, and Lagrange, at its close, had gone a step in this direction by saying that no series could safely be used unless one investigated the remainder.42 Nevertheless, no general definition, or theory, of convergent infinite series had been given, and Euler and Lacroix continued to employ divergent series in their work. With the turn of the next century, however, Abel, Bolzano, Cauchy, and Gauss all pointed out the need for definitions and tests of convergence of infinite series before the latter could legitimately be employed in mathematics. In this Ibid., pp. 151-52. » Saks, Théorie de l'intégrale, pp. 122-23. " See Cauchy, Œuvres (2), I, 335 and Œuvres (1), I, 335. 40 Saks, Théorie de l'intégrale, p. 125. ° See Vera circuii et hyperbolae quadratura, p. 10. u Reiff, Geschichte der unendlichen Reihen, pp. 69-70, 155.

17

The Rigorous Formulation

281

respect the work of Cauchy, in particular, laid the foundation of the theory of convergence and divergence through the wide influence which his work exerted upon his contemporaries. Cauchy defined a series as convergent if, for increasing values of n, the sum SH approaches indefinitely a certain limit S, the limit 5 in this case being called the sum of the series.43 Cauchy here showed clearly that the limit notion is involved, as it was also in differentiation and integration and in defining continuity. Furthermore, he pointed out that it is only in this sense that an infinite series may be regarded as having a sum. In other words, Zeno's paradox of the Achilles is to be answered in precisely such ideas, based upon the limit concept. Cauchy went on in this work to try to prove what has become known as Cauchy's theorem—that a necessary and sufficient condition that the sequence converge to a limit is that the difference between Sp and Sq for any values of p and q greater than n can be made less in absolute value than any assignable quantity by taking n sufficiently large. A sequence satisfying this condition is now said to converge within itself. The necessity of the condition follows immediately from the definition of convergence, but the proof of the sufficiency of the condition requires a previous definition of the system of real numbers, of which the supposed limit 5 is one. Without a definition of irrational numbers, this part of the proof is logically impossible. Cauchy had stated in his Cours d'analyse that irrational numbers are to be regarded as the limits of sequences of rational numbers. Since a limit is defined as a number to which the terms of the sequence approach in such a way that ultimately the difference between this number and the terms of the sequence can be made less than any given number, the existence of the irrational number depends, in the definition of limit, upon the known existence, and hence the prior definition, of the very quantity whose definition is being attempted. That is, one cannot define the number v 2 as the limit of the sequence 1, 1.4, 1.41, 1.414, . . . because to prove that this sequence has a limit one must assume, in view of the definitions of limit and convergence, the existence of this number as previously demonstrated or defined. Cauchy appears not to have noticed the circularity of the reasoning « Œuvres (2), III, 114.

282

The Rigorous Formulation

in this connection, 44 but tacitly assumed that every sequence converging within itself has a limit. That is, he felt that the existence of a number possessing the external relationship expressed in the definition of convergence and the sum of the series, would follow from the inner relations expressed in the Cauchy theorem. This idea may have been based upon the very thing that he and Bolzano had sought to avoid— that is, upon preconceptions taken over from geometry. The attempt to base the idea of number upon that of the geometrical line had given rise to the Pythagorean difficulty of the incommensurable and the ensuing development of the calculus. It had likewise suggested to Gregory of St. Vincent, two centuries before Cauchy, that the sum of an infinite geometrical progression could be represented by the length of a line segment and that the series could therefore be thought of as having a limit. However, in order to make the limit concept of analysis independent of geometry, mathematicians of the second half of the nineteenth century attempted to frame definitions of irrational number which did not make use of the definition of a limit. The geometrical intuitions which intruded themselves into Cauchy's view of irrational number likewise led him erroneously to believe that the continuity of a function was sufficient for its geometrical representation and for the existence of a derivative. 46 A. M . Ampère also had been led by geometric preconceptions similar to those of Cauchy to try to demonstrate the false proposition that every continuous function has a derivative, except for certain isolated values in the interval. 46 Bolzano had in his manuscripts of about this time given an example showing the falsity of such an opinion, but it remained for Weierstrass to make this fact known. With Cauchy, it may safely be said, the fundamental concepts of the calculus received a rigorous formulation. Cauchy has for this reason commonly been regarded as the founder of the exact differential calculus in the modern sense. 47 Upon a precise definition of the notion of limit, he built the theory of continuity and infinite series, of the derivative, the differential, and the integral. Through the popularity of his lectures and textbooks, his exposition of the calculus became Cf. Pringsheim, "Nombres irrationnels et notion de limite," p. 180. Cf. Jourdain, "The Theory of Functions with Cauchy and Gauss." 4 ' See Pringsheim, "Principes fondamentaux de la théorie des fonctions." 47 Klein, Elementary Mathematics from an Advanced Standpoint, p. 213. 44

The Rigorous Formulation

283

that generally adopted and the one which has been accepted down to the present time. Nevertheless, the use of the infinitely small persisted for some time. S. D. Poisson, in his Traité de mécanique which appeared in several editions in the first half of the nineteenth century and which was long a standard work, used exclusively the method of infinitesimals. These magnitudes, "less than any given magnitude of the same nature," he held to have a real existence. They were not simply "a means of investigation imagined by geometers." 48 The object of the differential calculus he consequently regarded as the determination of the ratio of infinitely small quantities, in which infinitesimals of higher order were neglected; 49 and the integral was the inverse of the differential quotient. A. A. Cournot likewise opposed the work of Cauchy, although upon somewhat different grounds. In his Traité élémentaire de la théorie des fonctions et du calcul infinitésimal of 1841, he asserted that his taste for the philosophy of science prepared him to treat the metaphysics of the calculus.60 In presenting this, his attitude resembled somewhat that of Carnot. He held that the theories of Newton and Leibniz complemented each other, and that the method of Lagrange represented simply a return to the views of Newton. 61 The infinitely small "existed in nature," as a mode of generation of magnitudes according to the law of continuity, although it could be defined only indirectly in terms of limits.62 However, Cournot protested that concepts exist in the understanding, independently of the definition which one gives to them. Simple ideas sometimes have complicated definitions, or even none. For this reason he felt that one should not subordinate the precision of such ideas as those of speed or the infinitely small to logical definition.63 This point of view is diametrically opposed to that which has dominated the mathematics of the last century. The tendency in analysis since the time of Cournot has been toward ever-greater care in the formal logical elaboration of the subject. This trend, initiated in the first half of the nineteenth century and fostered largely by Cauchy, was in the second half of that century continued with notable success by Weierstrass. " Train de mécanique, I, 13-14. m Traité élémentaire, Preface.

» Ibid.

52

Ibid., pp. 85-88.

« Ibid., pp. 14-16. « Ibid., p. 72.

284

The Rigorous Formulation

In spite of the care with which Cauchy worked, there were a number of phrases in his exposition which required further explanation. The expressions "approach indefinitely," " a s little as one wishes," "last ratios of infinitely small increments," were to be understood in terms of the method of limits, but they suggested difficulties which had been raised in the preceding century. The very idea of a variable approaching a limit called forth vague intuitions of motion and the generation of quantities. Furthermore, there were, in Cauchy's presentation, certain subtle logical gaps. One of these was the failure to make clear the notion of an infinite aggregate, which is basic in his work in infinite sequences, upon which the derivative and the integral are built. Another lacuna is evident in his omission of a clear definition of that most fundamental of all notions—number—which is absolutely essential to the definition of limits, and therefore to that of the concepts of the calculus. The first of these points had been touched upon by Bolzano, but the theory was not further developed until much later, largely through the efforts of Georg Cantor. In the second matter the difficulty is essentially that of a vicious circle in the definition of irrational numbers, and this Weierstrass sought to resolve. Although it was Cauchy who gave to the concepts of the calculus their present general form, based upon the limit concept, the last word on rigor had not been said, for it was Karl Weierstrass" who constructed a purely formal arithmetic basis for analysis, quite independent of all geometric intuition. Weierstrass in 1872 read a paper in which he showed what had been known to Bolzano sometime before—that a function which is continuous throughout an interval need not have a derivative at any point in this interval." Previously it had been generally held, upon the basis of physical experience, that a continuous curve necessarily possessed a tangent, except perhaps at certain isolated points. From this it would follow that the corresponding function should in general possess a derivative. Weierstrass, however, demonstrated conclusively the incorrectness of such suggestions M See Poincaré, "L'Œuvre mathématique de Weierstrass"; cf. also Pierpont, "Mathematical Rigor," pp. 34-36. " This seems to have been presented by Weierstrass in his lectures as early as 1861. See Pringsheim, "Principes fondamentaux," p. 45, n.; Voss, "Calcul différentiel," pp. 260-61,

The Rigorous Formulation

285

of experience. This he did by forming the nondifferentiable continm

uous function f(x) =

2 bacos(a\x),

#1-0

where x is a real variable, a an

odd integer, and b a positive constant less than unity such that ab > 1 + — 2

Since that time many other such functions have become known, and we may even say that, in spite of geometric intuition, of all continuous functions those with tangents at some points are the exceptions.47 Intuition has been even more discredited as a guide by the fact that we can have continuous curves defined by motion, which yet have no tangents." Inasmuch as it was apparent to Weierstrass that intuition could not be trusted, he sought to make the bases of his analysis as rigorously and precisely formal as possible. He did not present his work on the elements of the calculus in a number of treatises, as had Cauchy, nor even in a series of papers. His views became known, rather, through the work of students who attended his lectures.59 In order to secure logical exactitude, Weierstrass wished to establish the calculus (and the theory of functions) upon the concept of number alone,60 thus separating it completely from geometry. To do this it was necessary to give a definition of irrational number which should be independent of the limit idea, since the latter presupposes the former. Weierstrass was thus led to make profound investigations into the principles of arithmetic, particularly with respect to the theory of irrationals. In this work Weierstrass did not go into the nature of the whole number itself, but began with the concept of whole number as an aggregate of units enjoying one characteristic property in common, whereas a complex number was to be thought of as an aggregate of units of various species enjoying more than one char" See Weierstrass, Mathematische Werke, II, 71-74; Mansion, "Fonction continue sans derivie de Weierstrass." » Cf. Voss, "Calcul differentiel," pp. 261-62. " Neikirk, "A Class of Continuous Curves Defined by Motion Which Have No Tangents Lines." u See, for example, Pincherle, "Saggio di una introduzione alia teoria delle funzioni analitiche secondo i principii del Prof. C. Weierstrass." Cf. Merz, A History of European Thought in the Nineteenth Century, II, 703. M Jourdain, "The Development of the Theory of Transfinite Numbers," 1908-9, n., p. 298; cf. also p. 303.

286

The Rigorous Formulation

acteristic property. All rational numbers can then be defined by introducing convenient classes of complex numbers. Thus the number 3f is made up of 3 a and 2/3, where a is the principle unit and 0 is an aliquot part, taken as another element. A number is then said to be determined when we know of what elements (of which there is an infinite number) it is composed and the number of times each occurs. In this theory the number not defined as the limit of the sequence 1, 1.4, 1.41, . . . , nor is the idea of sequence brought in; it is simply the aggregate itself in any order l a , 40, I7, . . . where o is the principle unit and 0, y, . . . are certain of its aliquot parts, and where the aggregate is, of course, subject to the condition that the sum of any finite number of elements is always less than a certain rational number. We can now prove, if we wish, that this number is the limit of the variable sequence l a ; la, 4/3; la, 40, I7; . . . , thus correcting the logical error arising in Cauchy's theory of number and limits.01 In a sense, Weierstrass settles the question of the existence of a limit of a convergent sequence by making the sequence (really he considers an unordered aggregate) itself the number or limit. In making the basis of the calculus more rigorously formal, Weierstrass also attacked the appeal to intuition of continuous motion which is implied in Cauchy's expression—that a variable approaches a limit. Previous writers generally had defined a variable as a quantity or magnitude which is not constant; but since the time of Weierstrass it has been recognized that the ideas of variable and limit are not essentially phoronomic, but involve purely static considerations. Weierstrass interpreted a variable x as simply a letter designating any one of a collection of numerical values.62 A continuous variable was likewise defined in terms of static considerations: If for any value x0 of the set and for any sequence of positive numbers ..., {„, however small, there are in the intervals x0 — §,, x„ -f 5, others of the set, this is called continuous. 63 Similarly, for a continuous function Weierstrass gave a definition equivalent to those of Bolzano and Cauchy, but having greater clarity and precision. To say that ]{x + Ax) — fix) becomes infinitesimal, tl

Ibid., 1908-9, pp. 303 ff. Cf. Russell, Principles of Mathematics, pp. 281 ff.; Pringsheim, "Nombres irrationnels et notion de limite," pp. 149 ff.; Pincherle, op. cit., pp. 179 ff. M « Pincherle, op. cit., p. 234. Ibid., p. 236.

The Rigorous Formulation

287

or becomes and remains less than any given quantity, as Ax approaches zero, calls to mind either the infinitely small or else vague notions of mobility. Weierstrass defined f(x) as continuous, within certain limits of x, if for any value x„ in this interval and for an arbitrarily small positive number it is possible to find an interval about x0 such that for all values in this interval the difference f(x) — f(x0) is in absolute value less than e ; M or, as Heine was led by the lectures of Weierstrass to express it, if, given any «, an v„ can be found such that for f < v0, the difference fix ± v) — f(x) is less in absolute value than t 65 The limit of a variable or function is similarly defined. The number L is the limit of the function fix) for x = x0 if, given any arbitrarily small number e, another number 5 can be found such that for all values of x differing from x0 by less than 8, the value of fix) will differ from that of L by less than t-66 This expression of the limit idea, in conjunction with Cauchy's definitions of the derivative and the integral, supplied the fundamental conceptions of the calculus with a precision which may be regarded as constituting their rigorous formulation. There is in this definition no reference to infinitesimals, so that the designation "the infinitesimal calculus," which is used even today, is shown to be inappropriate. Although a number of mathematicians, from the time of Newton and Leibniz to that of Bolzano and Cauchy, had sought to avoid the use of infinitely small quantities, the unequivocal symbolism of Weierstrass may be regarded as effectively banishing from the calculus the persistent notion of the fixed infinitesimal. During the eighteenth century there had been a lively argument, both in connection with the prime and ultimate ratio of Newton and with the differential quotient of Leibniz, as to whether a variable which approaches its limit can ever attain it. This is essentially the crux of Zeno's argument in the Achilles. In the light of the precision of the Weierstrassian theory of limits, however, the question is seen to be entirely inapposite. The limit concept does not involve the idea of approaching, but only a static state of affairs. The single question "Ibid., p. 246. "Heine, "Die Elemente der Funktionenlehre," p. 182; cf. also p. 186. M See Stolz, Vorlesungen uber allgemeine Arithmetik, I, 156-57; cf. also Whitehead, Introduction to Mathematics, pp. 226-29.

An

288

The Rigorous Formulation

amounts really to two: first, does the variable f(x) have a limit L for the value a of x. Secondly, is this limit L the value of the function for the value a of x. If /(a) = L, then one can say that the limit of the variable for the value of x in question is the value of the variable for this value of x, but not that f(x) reaches f(a) or L, for this latter statement has no meaning. In retrospect, it is pertinent to remark that whereas the idea of variability had been banned from Greek mathematics because it led to Zeno's paradoxes, it was precisely this concept which, revived in the later Middle Ages and represented geometrically, led in the seventeenth century to the calculus. Nevertheless, as the culmination of almost two centuries of discussion as to the basis of the new analysis, the very aspect which had led to its rise was in a sense again excluded from mathematics with the so-called "static" theory of the variable which Weicrstrass had developed. The variable does not represent a progressive passage through all the values of an interval, but the disjunctive assumption of any one of the values in the interval. Our vague intuition of motion, although remarkably fruitful in having suggested the investigations which produced the calculus, was found, in the light of further elaboration in thought, to be quite inadequate and misleading. What, then, about that obscure and elusive feeling for continuity which colors so much of our thought? Is that baseless also? What about the idea of infinity upon which Bolzano had speculated and which Weierstrass had used, somewhat covertly, in his definition of irrational number? Can this be given a consistent definition? These questions were investigated largely by Dedekind and Cantor, two mathematicians who were thinking along lines similar to those which Weierstrass had followed in seeking a satisfactory definition of irrational numbers. The year 1872 was, for a number of reasons, a significant one in the history of the foundations of the calculus. I t saw, besides the presentation by Weierstrass of his continuous nondifferentiable function and the publication by one of his students of Weierstrass' lectures on the elements of arithmetic, 67 the appearance of the following: Nouveau precis d'analyse infinilesimale of Charles Meray; a paper in Crelle's Journal by Eduard Heine on "Die Elemente der Funktionenlehre"; 87 Kossak, Die Elemente der ArithmeUk.

The Rigorous Formulation

289

the first paper by Georg Cantor on the principles of arithmetic, which appeared in the Mathematische Annalen as "Über die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen"; and the Stetigkeit und die Irrationalzahlen of Richard Dedekind. Incidentally the work of each of these men touched upon one and the same problem—that of formulating a definition of irrational number which should be independent of that of the limit concept." The work of Weierstrass in this connection has already been described. With respect to publication, this had been anticipated by Méray, who in 1869, in an article entitled "Remarques sur la nature des quantités définies par la condition de servir de limites à des variables données," sought to resolve the vicious circle in the definitions of limit and irrational number given by Cauchy. Three years later Méray further elaborated his views, in his Nouveau précis. It will be recalled that Bolzano and Cauchy had attempted to prove that a sequence which converges within itself—that is, one for which, given any e, however small, an integer N can be found such that for n > N and for any integral value of p greater than the integer n, the inequality |5„ + p — S j < t will hold—converges in the sense of external relations, that is, that it has a limit 5. Méray, in this connection, cut the Gordian knot by rejecting Cauchy's definition of convergence in terms of the limit S. He called an infinite series convergent if it converged within itself, according to Cauchy's theorem. In this case one need not demonstrate the existence of an undefined number S, which may be regarded as the limit. The word number in the strict sense Méray reserved for the integers and rational fractions; the converging sequence of rational numbers, which Méray called a convergent "variante," he regarded as determining a number in the broad sense, rational or irrational. He was somewhat vague as to whether or not the sequence is the number.69 If so, as is implied in the case of irrationals, his theory is equivalent to that of Weierstrass, although somewhat less explicitly expressed. Attempts, similar to those of Weierstrass and Méray, to avoid the M For a full account of this work, together with bibliographical references, see Jourdain, "The Development of the Theory of Transfinite Numbers"; see also Pringsheim, "Nombres irrationnels et notion de limite," pp. 144 ff. • •• See "Remarques sur la nature des quantités"; also Nouveau précis, pp. xv, 1-7; cf. also Jourdain, "The Development of the Theory of Transfinite Numbers,'* 1910, pp. 28 ff.

290

The Rigorous Formulation

pelitio principii in Cauchy's reasoning on limits and irrational numbers were developed and published, also in 1872, by Cantor and by Heine. Meray had avoided the logical difficulty by taking + p — < € as the definition of convergence, instead of the condition |S — Sn| < €, which presupposes the demonstrated existence of S. Likewise in Weierstrass' definition the irrational numbers are expressed in an analogous manner, not as limits but as entire infinite groups of rational numbers. The work of these men led Heine and Cantor to express similar views. Rather than postulate the existence of a number S, which is the limit of an infinite series which converges within itself, they considered S, not exactly as determined by the series, as Meray had somewhat indecisively held, but as defined by the series—as simply a symbol for the series itself.70 Their definitions resemble the view of Weierstrass, with the addition of the condition of Meray that ^ ^ (Sn + p — SH) = 0 for p arbitrary. This condition is equivalent to that of Weierstrass—whose aggregates were such that however summed in finite number, the sum was to remain below a certain limit—but is expressed in a rather more convenient form. Still another attempt along these lines was made by Dedekind. Weierstrass had been lecturing on the theory of functions in 1859 and had in this way been led to investigations concerning the foundations of arithmetic. In like fashion Dedekind admitted that his attention was directed to these matters when in 1858 he found himself obliged to lecture for the first time on the elements of the differential calculus.71 In discussing the notion of the approach of a variable magnitude to a fixed limiting value, he had recourse, as had Cauchy before him, to the evidence of the geometry of continuous magnitude. He felt, however, that the theory of irrational numbers, which lay at the root of the difficulty in the limit concept, should be developed out of arithmetic alone, if it were to be rigorous.72 Dedekind's approach to the problem was somewhat different from that of Weierstrass, Meray, Heine, and Cantor in that, instead of 70 Heine, "Die F.lemente der Funktionenlehre," pp. 174 ff.; Cantor, Gesammelte Abhandlungen, pp. 92-102, 185-86. See also the articles by Jourdain, " T h e Development of the Theory of Transfinite Numbers," 1910, pp. 21-43, and "The Introduction of Irrational Numbers." Cf. Russell, The Principles of Mathematics, pp. 283-86. 71 Dedekind, Essays on the Theory of Numbers, p. 1. n Ibid., pp. 1-3; cf. also p. 10.

The Rigorous Formulation

291

considering in what manner the irrationals are to be defined so as to avoid the vicious circle of Cauchy, he asked himself what there is in continuous geometrical magnitude which resolved the difficulty when arithmetic apparently had failed: i. e., what is the nature of continuity? Plato had sought to find this in a vague flowing of magnitudes; Aristotle had felt that it lay in the fact that the extremities of two successive parts were coincident. Galileo had suggested that it was the result of an actually infinite subdivision—that the continuity of a fluid was in this respect to be contrasted with the finite, discontinuous subdivision illustrated by a fine powder. The philosophy and mathematics of Leibniz had led him to agree with Galileo that continuity was a property concerning disjunctive aggregation, rather than a unity or coincidence of parts. Leibniz had regarded a set as forming a continuum if between any two elements there was always another element of the set.73 The scientist Ernst Mach likewise regarded this property of the denseness of an assemblage as constituting its continuity, 74 but the study of the real number system brought out the inadequacy of this condition. The rational numbers, for example, possess the property of denseness and yet do not constitute a continuum. Dedekind, thinking along these lines, found the essence of the continuity of a line to be brought out, not by a vague hang-togetherness, but in the nature of the division of the line by a point. He saw that in any division of the points of a line into two classes such that every point of the one is to the left of every point of the other, there is one and only one point which produces this division. This is not true of the ordered system of rational numbers. This, then, was why the points of a Une formed a continuum, but the rational numbers did not. As Dedekind expressed it, "By this commonplace remark the secret of continuity i§ to be revealed." 75 It is obvious, then, in what way the domain of rational numbers is to be rendered complete to form a continuous domain. It is only necessary to assume the Cantor-Dedekind axiom that the points of a line can be put into one-to-one correspondence with the real numbers. Arithmetically expressed, this means that for every division of the 73 74

Philosopkische Schrijten, II, 515. Die Principien der Wármelehre, p. 71.

7S

Essays on the Theory of Numbers, p. 11.

292

The Rigorous Formulation

rational numbers into two classes such that every number of the first, A, is less than every number of the second, B, there is one and only one real number producing this Schnitt, or "Dedekind Cut." Thus if we divide the rational numbers into two classes A and B, such that A contains all those whose squares are less than two and B all those whose squares are more than two, there is, by this axiom of continuity, a single real number—written in this case as which produces this division. Furthermore, this cut constitutes the definition of the number Similarly, any real number is defined by such a cut in the rational number system. This postulate makes the domain of real numbers continuous, in the sense that the straight line has this property. Moreover, the real number of Dedekind is in a sense a creation of the human mind, independent of intuitions of space and time. The calculus had been generally recognized as dealing with continuous magnitude, but before this time no one had explained precisely the sense in which this was to be accepted. Symbols for variables had displaced the idea of geometrical magnitude, but Cauchy implied a geometrical interpretation of a continuous variable. Dedekind showed that it was not, as had frequently been held,76 the apparent freedom from the discreteness of the rational numbers which made geometrical quantities continuous, but only the fact that the points in them formed a dense, perfect set. On completing the number system in the manner suggested by this fact—that is, by adopting Dedekind's postulate—this system was made continuous also. Now the fundamental theorems on limits could be proved rigorously77 and without recourse to geometry, as Dedekind pointed out, on the basis of his new definition of real number. 78 Geometry having pointed the way to a suitable definition of continuity, it was in the end excluded from the formal arithmetical definition of this concept. The Dedekind Cut is in a sense equivalent to the definitions of real number given by Weierstrass, Meray, Heine, and Cantor. 79 Bertrand ™ See Drobisch, "Ueber den Begriff des Stetigen und seine Beziehungen zum Calcul," p. 170. 77 It should be noted, however, that Dedekind's definition of number has recently been criticized as involving a vicious circle. See Weyl, "Der Circulus vitiosus in der heutigen Begründung der Analysis." 78 Essays on the Theory of Numbers, p. 27; cf. also pp. 35-36. 79 See J. Tannery, Review of Dantscher, Vorlesungen über die Weierstrassche Theorie der irrationalen Zahlen.

The Rigorous Formulation

293

Russell followed the line of thought suggested by these men, in attempting another formal definition of real number. He felt that the definitions previously given either disregarded the question of the existence of the irrational numbers or artificially postulated new numbers, leaving some doubt as to just what they are. He suggested that a real number be defined as a whole "segment" of the rational numbers. The number V ^ for example, is defined as the ordered aggregate of all rational numbers whose squares are less than two. That is, instead of postulating an element dividing the rational numbers into two classes, as Dedekind did, he would merely take one of Dedekind's classes and make it, rather than the cutting element, the number.*0 This obviates the necessity of introducing any conception other than that of rational number and segment of rational numbers. According to this view, there is no need to create the irrational numbers; they are at hand in the system of rational numbers, as they had been also in the somewhat more involved doctrine of Weierstrass. The object of all the above efforts in the establishment of the real number was to give a formal logical definition which should be independent of the implications of geometry and which should avoid the logical error of defining irrational numbers in terms of limits, and vice versa. From these definitions, then, the basic theorems on limits in the calculus can be derived without circularity in reasoning. The derivative and the integral are thus established directly on these definitions, and are consequently divested of any character connected with sensory perception, such as rate of change or surface area. Geometrical conceptions cannot be made sufficiently explicit and precise, as we have seen during our consideration of the long period of development of the calculus. Thus the required rigor was found in the application of the concept of number, made formal by divorcing it from the idea of geometrical quantity. From the definitions of number given above, we see that it is not magnitude which is basic, but order. This is brought out most clearly in the definitions given by Dedekind and Russell, these involving only ordered classes of elements. The same is true, however, of the other systems, which have the disadvantage of requiring new definitions of equality before this can be made clear. The essential characteristic of the number two is not its *> Russell, Introduction to Mathematical Philosophy, p. 72.

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The Rigorous Formulation

magnitude, but its place in the ordered aggregate of real numbers. The derivative and the integral, although still defined as limits of characteristic quotients and sums respectively, have, as a result, ultimately become, through the definition of number and limit, not quantitative but ordinal concepts. The calculus is not a branch of the science of quantity, but of the logic of relations. Dedekind's work not only met the need for a definition of number independent of that of limit, but in addition gave an explanation of the nature of continuous magnitude. Bolzano, Cauchy, and others had given definitions of a continuous function of an independent variable. A continuous, independent variable was tacitly understood as one which could take on all values in an interval corresponding to the points of a line segment. The arithmetization of 1872, however, went beyond the geometrical picture and expressed formally, in terms of ordered aggregates, what was meant by a continuous variable or ensemble. The conditions were: first, that the values or elements should form an ordered set; second, that this should be a dense set— that is, between any two values or elements, there should always be others; and third, that the set should be perfect—that is, if the elements are divided, as in a Dedekind Cut, there should always be one which produces this cut. This definition is far removed from any appeal to empiricism and from the picture of a smooth, unbroken "oneness" or cohesiveness, which instinctive feeling associates with the notion of continuity. I t specifies only an infinite, discrete multiplicity of elements, satisfying certain conditions—that the set be ordered, dense, and perfect. This is the sense in which one is to interpret the remark that the calculus deals with continuous variables; the sense in which one is to interpret Newton's phrase "prime and ultimate ratios," or the ultimate relationship between the differentials which Leibniz thought subsisted by virtue of the law of continuity. The introduction of uniform motion into Newton's method of fluxions was an irrelevant evasion of the question of continuity, disguised by an appeal to intuition. There is nothing dynamic in the idea of continuity, nor, so far as we know, is the converse necessarily true. By sense perception we are apparently unable to conclude whether or not we are dealing in motion with a continuum. The experiments of Helmholtz, Mach, and others have

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shown that the physiological spaces of touch and sight are themselves discontinuous.81 The continuity of time which Barrow and Newton regarded as assured by its relentless even flow is now seen to be simply a hypothesis. Mathematics is unable to specify whether motion is continuous, for it deals merely with hypothetical relations and can make its variable continuous or discontinuous at will. The paradoxes of Zeno are consequences of the failure to appreciate this fact and of the resulting lack of a precise specification of the problem. The dynamic intuition of motion is confused with the static concept of continuity. The former is a matter of scientific description a posteriori, whereas the latter is a matter solely of mathematical definition a priori. The former may consequently suggest that motion be defined mathematically in terms of continuous variables, but cannot, because of the limitations of sensory perception, prove that it must be so defined. If the paradoxes of Zeno are thus stated in the precise mathematical terminology of continuous variables and of the derived concepts of limit, derivative, and integral, the seeming contradictions resolve themselves. The dichotomy and the Achilles depend upon the question as to whether or not the sets involved are perfect.82 The stade is answered upon the basis of dense sets, and the arrow by the definition of instantaneous velocity, or the derivative. The mathematical theory of continuity is based, not on intuition, but on the logically developed theories of number and sets of points. The latter, however, depend, in turn, upon the idea of an infinite aggregate, an idea which Zeno had invoked to fortify his arguments. Zeno's appeal to the infinite was based upon the supposed inconceivability of the notion of completing in a finite time an infinite number of steps. It is again the scientific description a posteriori which he questioned, but, so far as we know, there is no way of proving or disproving the possibility, not only of the existence of infinite aggregates in the physical sense, but also of the execution in thought of an infinite number of steps in connection with aggregates, whether finite or infinite. Since science cannot answer this point, the question may become a hypothetical mathematical one. " Enriques, Problems of Science, pp. 211-12. 82 Cf. Helmholtz, Counting and Measuring, p. xviii; Cajori, "History of Zeno's Arguments on Motion," p. 218.

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The Rigorous Formulation

Mathematics, moreover, requires a theory of the infinite in its definitions of number and continuity. The question of the mathematical existence, i. e., of the consistent logical definition, of infinite aggregates therefore remained to be answered. Galileo had suggested vaguely, and Bolzano had seen more clearly, that infinite ensembles must have the paradoxical property that a part can be put into oneto-one correspondence with the whole. This fact led Cauchy to deny their existence; and indeed upon the basis of the work of Cauchy and Weierstrass, one could have said that the infinite indicated nothing more than the potentiality of Aristotle—an incompleteness of the process in question. 83 Their infinitesimals were variables having zero as their limit, and the limit concept involved only the definition of number. However, this very definition of number implicitly presupposes the prior existence of infinite aggregates, so that this question could not indefinitely be avoided. Under the influence of Weierstrass' work in the foundations of arithmetic, Dedekind and Cantor sought a basis for the theory of infinite aggregates, 84 in order to complete this work. This they found in Bolzano's paradox. Instead of looking upon it as merely a strange property of infinite aggregates, they made it the definition of an infinite set. Dedekind said "A system 5 is said to be infinite when it is similar to a proper part of itself; in the contrary case 5 is said to be a finite system."8® Under this definition, infinite aggregates exist as logically self-consistent entities, and the definitions of real number are completed. Cantor, with whom Dedekind corresponded in this connection, 86 was not satisfied with merely defining infinite sets. He wished to develop the subject further. In a series of papers he reviewed the history of the infinite from the time of Democritus to that of Dedekind, and elaborated his theory of infinite ensembles, or Mengenlehre. Cantor's doctrine of the mathematical infinite, which has been hyperbolized as "the only genuine mathematics since the Greeks," 87 did not refer to the potential infinity of Aristotle nor the syncategorematic infinity of the Scholastics. These were bound up always with variability. 88 Cantor referred instead to the categorematic infinity of medieval w See Hilbert, "Über das Unendliche." ° See Baumann, "Dedekind und Bolzano." Dedekind, Essays on the Theory of Numbers, p. 63. M See Georg Cantor, Gesammelte Abhandlungen. " See Bell, The Queen of the Sciences, p. 104. • Ibid., p. 180.

M

The Rigorous Formulation

297

philosophy—the eigentlich- Unendlichen. He felt, with some justice, that Scholastic thought had handled this subject more as a religious dogma than as a mathematical concept." Moreover, the thought of Leibniz, whose calculus represented the most genial attempt to establish a mathematics of the infinitely large and the infinitely small, lacked resolution. Sometimes he declared against the absolute infinite, and then again he remarked that nature, instead of abhorring the actual infinite, everywhere made use of it to mark better the perfections of its author. 90 The symbol °° had been used by mathematicians since the time of Wallis to represent infinity, but no definition had been given, nor had the Scholastic distinction been observed. The symbol was used by Weierstrass, for example, in the sense both of a potentiality and of an actuality: he wrote /(a) = °° to mean that

= 0, and also f w used the expression / ( 0 0 ) = b in the sense that the limit of f(x) for x indefinitely large was b.91 To avoid this confusion, Cantor chose a new symbol, a>, to represent the actual infinite aggregate of positive integers. It is to be remarked, furthermore, that whereas o° referred in general to magnitude, u> is to be interpreted in terms of aggregation. Wallis, Fontenelle, and others regarded oo variously as the largest positive integer or as the sum of all the positive integers; but the symbol a refers to all the positive integers only in the sense that these form an aggregate of elements. This view of the infinite, as concerned with groups of elements, had previously been clearly expressed by Bolzano, but he failed to recognize what Cantor called the power of an infinite set of elements. The rational numbers can be put into one-to-one correspondence with the positive integers, and for this reason these two classes are regarded as having the same power. One of the most striking results of Cantor's Mengenlehre, however, is that there are transfinite numbers higher than The theory of arithmetic had shown that numbers other than the rationals were needed for the continuum, and Cantor now showed that the desired set of real numbers was such that these could not be put into one-toone correspondence with the positive integers; i. e., the set was not " Ibid., p. 191. *> Ibid., p. 179; see also Leibniz, Philosophised Schriften, I, 416. M Cajori, A History of Mathematical Notations, II, 45.

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The Rigorous Formulation

denumerable. It therefore represented a transfinite number of a higher power, which is often written now simply as C. Other numbers have been found above C, but the question as to whether there is one between u and C is unanswered. However, the definition of the continuous variable, and hence of the concepts of the calculus, requires only the infinite set C. Although Cantor's work does not clear up any objection raised on the ground of the conceptual difficulties inherent in the concept of infinity, it definitely refutes any argument raised upon the score of logical contradiction. Likewise, any criticism of the use of the infinite in defining irrational number or in the limit concept is answered by Cantor's work, which clarifies the situation. 92 As the terminus a quo of the investigations leading to the calculus is to be found in the Pythagorean discovery of the incommensurable and in the recognized need for satisfactory definitions of number and the infinite, so the terminus ad quern may be regarded as the establishment of these by the great triumvirate: Weierstrass, Dedekind, and Cantor. The fundamental notions of the calculus—the limit of a continuous variable: the derivative and the integral—have through the work of these men been given a logical rigor as impressive as that of Euclidean geometry, and a formal precision of which the Greeks had never dreamed. It has been shown that in analysis there is no need for anything but whole numbers, or finite or infinite systems of these. 93 How startlingly apropos, with respect to the development of the calculus, is the Pythagorean dictum: All is number! " It should, perhaps, be observed at this point that the theory of infinite aggregates has resulted also in a number of puzzling and as yet unresolved antimonies. See Poincaré, Foundations of Science, pp. 477 fit.; Pierpont, "Mathematical Rigor," pp. 42-44. It has been suggested, in this connection, that these paradoxes may be related to the difficulties encountered in theoretical physics. See the review by Northrop, in Bulletin, American Mathematical Society, X L I I (1936), 791-92, of Schrodinger, Science and the Human Temperament. ™ Poincaré, The Foundations of Science, pp. 380, 441 5.

VIII. Conclusion

T

H E R E is a strong temptation on the part of professional mathematicians and scientists to seek always to ascribe great discoveries and inventions to single individuals. Such ascription serves a didactic end in centering attention upon certain fundamental aspects of the subjects, much as the history of events is conveniently divided into epochs for purposes of exposition. There is in such attributions and divisions, however, the serious danger that too great a significance will be attached to them. Rarely—perhaps never—is a single mathematician or scientist entitled to receive the full credit for an "innovation," nor does any one age deserve to be called the "renaissance" of an aspect of culture. Back of any discovery or invention there is invariably to be found an evolutionary development of ideas making its geniture possible. The history of the calculus furnishes a remarkably apt illustration of this fact. 1 The method of fluxions of Newton was no more unanticipated than were his laws of motion and gravitation; and the differential calculus of Leibniz had been as fully adumbrated as had his law of continuity. These two men are to be thought of as the inventors of the calculus in the sense that they gave to the infinitesimal procedures of their predecessors the algorithmic unity and precision necessary for further development. Their work differed from the corresponding methods of their predecessors, Barrow and Fermat, more in attitude and generality than in substance and detail. The procedures of Barrow and Fermat were themselves but elaborations of the views of such men as Torricelli, Cavalieri, and Galileo, or Kepler, Valerio, and Stevin. The achievements of these early inventors of infinitesimal devices were in turn the direct results of the contributions of Oresme, Calculator, Archimedes, and Eudoxus. Finally, the work of the last-named men was inspired by the mathematical and philosophical problems suggested by Aristotle, Plato, Zeno, and Pythagoras. Without the filiation of ideas which was built up by these men and many others, the calculus of Newton and Leibniz would be unthinkable. 1

Cf. Karpinski, "Is there Progress in Mathematical Discovery?" pp. 47-48.

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If, on the one hand, mathematicians have been prone to forget the periods of suggestion and anticipation in the rise of the calculus, historians of the subject, on the other hand, have frequently failed to appreciate the significance of the later rigorous formulations. Historical accounts of the subject all too often have terminated with the work of Newton and Leibniz, even though neither of these men was able to furnish the precision of thought which was to follow two centuries later. The neglect of this later period of investigation indicates an inadequate attention to, or appreciation of, the fundamental concepts of the subject as presented by Cauchy and Weierstrass. References, which might easily have been multiplied, have frequently been indicated above, in which the views expressed in the final elaboration of the calculus have most unwarrantedly been imputed to earlier investigators in the field. Weierstrass' definition of real number has been identified with the theory of proportion of Eudoxus, and the Dedekind Cut with the speculations of Bryson. The continuum of Cantor has been viewed as expressed in the speculations of William of Occam or of Zeno. The limit concept of Weierstrass has been interpreted as identical with the prime and ultimate ratio of Newton, or even with the ancient Greek method of exhaustion. The derivative and the differential of Cauchy have been described as exactly corresponding to the related conceptions of Leibniz. The definite integral of Cauchy has been ascribed in all completeness to Fermat, or even to Cavalieri and Archimedes. Such citations make clear how general is the tendency unguardedly to read into the minds of earlier men one's own clear thoughts on the subject, forgetting that these are the culmination of centuries of speculation and investigation. The concepts of mathematics and science are eminently cumulative in their growth—the results of a continuous effort to understand the relationships between elements and, in terms of these, to describe the confused impressions afforded by physical experience. The dynamics and astronomy of the sixteenth century, for example, were not entirely new developments, but rather grew out of medieval and ancient views on these subjects. In the same sense the use of infinitesimal conceptions during the early modern period did not proceed de novo, b u t began where the Scholastic philosophers and Greek mathematicians had left off.

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Nevertheless, the fact of such progressive achievement is not to be interpreted as the unfolding of a well-conceived plan. Throughout the advance of science and mathematics, elements have constantly been discarded as well as added. For this reason no investigator is able to foresee the direction which the elaboration of his views will take. Only in retrospect can one trace the path along which such development has proceeded. Although in retracing this thread of thought one can readily recognize the notions from which the ultimate concepts have sprung, the former are not in general to be identified with the latter. Each is to be considered in the light of the mathematical and scientific milieu of the period in which it appeared. To interpret the geometric views of Archimedes and Barrow, for example, in terms of the analytical symbolism of the twentieth century is tantamount to invoking implicitly the precision and economy of thought which modern notations afford but which these earlier investigators were far from possessing. A deeper and more sympathetic understanding among professional workers in the fields of mathematics and history might easily remove much of the misdirected thought with respect to the nature and rise of mathematical concepts. A familiarity not only with the elements of the calculus, but also with the history of its development, will serve to bring out that the question is not so much who are the founders of the subject—Weierstrass and Cauchy, or Newton and Leibniz, or Barrow and Fermat, or Cavalieri and Kepler, or Archimedes and Eudoxus—but rather in what sense each of these men may be regarded as responsible for the new analysis. It is possible not only to trace the path of development throughout the twenty-five-hundred-year interval during which the ideas of the calculus were being formulated, but also to indicate certain tendencies inimical to this growth. Perhaps the most manifest deterring force was the rigid insistence on the exclusion from mathematics of any idea not at the time allowing of strict logical interpretation. The very concepts which gave birth to the calculus—those of variation and continuity, of the infinite and the infinitesimal—were banned from Greek mathematics for this reason, the work of Euclid being a monument to this exclusion. The work of Archimedes became most fruitful when he abandoned the Greek logical ideal and applied such forbidden

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concepts, but this represents an almost isolated example of their use. Similarly, in the seventeenth century a number of mathematicians, including Pascal and Barrow, avoided the use of algebra and analytic geometry as not compatible with the demands of rigor inherited from the ancient Greeks. Had they made full use of these elements, they might well have been acclaimed the inventors of the calculus. Again in the eighteenth century English mathematicians disdained, largely because of the weak logical basis (as well as for reasons of national jealousy), to use the differential method of the Continental "computists," and consequently failed to make appreciable contributions to the rapid growth of analysis characterizing that century. I t is clear that the indiscriminate use of methods and ideas which are palpably without logical foundation is not to be condoned. Such logical basis is, of course, ultimately to be sought in order to avoid hopeless confusion (as witness the eighteenth-century use of infinite series); but pending the final establishment of this, the banishment of suggestive views is a serious mistake. On the other hand, perhaps a more subtle, and therefore serious, hindrance to the development of the calculus was the failure, at various stages, to give to the concepts employed as concise and formal a definition as was possible at the time. The paradoxes of Zeno are excellent illustrations of the obscurity which results from a failure to specify clearly and unambiguously the conditions of the problem and to give formal definitions of the terms involved. Had the Greeks demanded of Zeno a precision of statement which the mathematicians exacted of themselves, they might not have banned the concepts leading to the calculus nor disregarded almost entirely the science of dynamics. The nice distinctions of the Scholastic philosophers pointed the way to the clarification of such problems, but these men were not sufficiently familiar with the formalism of Greek geometry and Arabic algebra to be able to carry their ideas to completion. With the decline of Scholasticism, the tendency was away from precision of thought and toward the free use of imagination, as found in the literary Renaissance. The mathematical counterpart of this is seen strongly in the works of Nicholas of Cusa and Kepler, who employed in mathematics, without seeking adequately to define them, the conceptions of infinity, the infinitesimal, motion, and continuity.

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In a sense this was fortunate, in that it favored the development of methods anticipating the calculus. On the other hand, however, the lack of a sound critical attitude was to leave undefined for hundreds of years the logical bases of the procedures thus employed, while the resulting confusion of thought engendered half a dozen alternative methods. Had Newton been more precise in the statement of his limit method, and had Leibniz been more explicit in professing that he was developing an instrument of invention and not a logical foundation, the period of indecision might not have ensued. As it was, it required the work of Cauchy, Weierstrass, and others to impart to the concepts of the continuous variable, the limit, the derivative, and the integral a precision of formulation which made them generally acceptable. In all probability, however, the chief obstacle in the way of the development of the concepts of the calculus was a misunderstanding as to the nature of mathematics. Ever since the empirical mathematics of the pre-Hellenic world was developed, the attitude has, upon occasion, been maintained that mathematics is a branch either of empirical science or of transcendental philosophy. In either case mathematics is not free to develop as it will, but is bound by certain restrictions: by conceptions derived either a posteriori from natural science, or assumed to be imposed a priori by an absolutistic philosophy. At the Egyptian and Babylonian level, mathematics was largely a body of information concerning the natural world. The early Ionians rearranged this knowledge into a deductive scheme, but the basis was still largely empirical science. The oriental mysticism of Pythagoras, however, reversed this state of affairs and gave to mathematics a supra-sensuous reality, of which the world of appearances was a counterpart. The premises were thus categorically established and all the mathematician could do was to develop the logical implications of these. This view was elaborated by Plato into an idealistic philosophy which has consistently denied the purely logical and hypothetical nature of the propositions of mathematics. At the Greek stage of the ideas leading to the calculus, however, the views of Plato exerted a favorable influence, in that they counterbalanced the Peripatetic attitude. Aristotle considered mathematics an idealized abstraction from natural science, and as such the premises and definitions were not

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arbitrary, but were determined by our interpretation of the world of sense perception. The concepts allowed to geometry were only such as were consistent with this picture. The infinitesimal and the actual infinite were excluded, not so much because of any demonstrated logical inconsistency, but because of a supposed incompatibility with the world of nature, from which the entities of mathematics were regarded as derived by a disassociation of irrelevant properties. Continuity depended upon coincidence of extremities and upon oneness, as sense perception appeared to indicate. Likewise number was regarded, in conformity with the judgments of empiricism and common sense, as a collection of units, with the result that irrational magnitudes were not considered as pertaining to the realm of number. Mathematics was the logic of relations, but the nature of these was completely determined by postulates which were in turn dictated by the evidence of physical experience. Of these two views, the Platonic and the Aristotelian, the former was for a time that under which the ideas of the calculus developed. Under this view, the conceptions of the infinite and the infinitesimal were not excluded, inasmuch as reason was not subject to the world of sensation. The entities of mathematics had an ontological reality, independent of common sense, and the postulates were discovered by reason alone. Although this made mathematics independent of natural science, it did not give it the postulational freedom it enjoys today. While the work of Archimedes displays elements of both the Platonic and the Aristotelian views, it was the latter which triumphed in the method of exhaustion and the classic geometry of Euclid. The concepts of infinity and continuity were consequently, during the medieval period, discussed from a dialectical rather than a mathematical point of view; but when they entered into the geometry of Nicholas of Cusa, of Kepler, of Galileo, it was largely under the Platonic view of rational transcendentalism, rather than of naturalistic description. Nevertheless, other mathematicians—Roberval, Torricelli, Barrow, and Newton—were led by the natural science of the day to interpret mathematics in terms of sense perception, as Aristotle had, and to introduce motion to avoid the difficulties of the infinite and the continuous. The philosophers Hobbes and Berkeley felt the empirical tendency so strongly that they denied to mathematics the idealized

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concept of a point without extension, because they felt that this had no counterpart in nature. The attitude of most of the mathematicians of the seventeenth century, however, was that of doubt. They employed infinitesimals and the infinite on the assumption that they existed, and treated the continuous as though made up of indivisibles, the results being justified pragmatically by their consistency with Euclidean geometry. In any case, the attitude was not that of an unprejudiced postulation and definition, followed by logical deduction. The investigations into the foundations of the calculus consequently took the form, during the eighteenth century, of a search for an explanation which should be intuitively plausible, rather than logically self-consistent. At this time, however, there was rapidly developing a very successful algebraic formalism, vigorously fostered by Euler and Lagrange. This led, in the nineteenth century, to a view of mathematics which nonEuclidean geometry had strongly suggested—a postulational system independent alike of the world of sense experience and of any dictates resulting from introspection. The calculus became free to adopt its own premises and to frame its own definitions, subject only to the requirement of an inner consistency. The existence of a concept depended only upon a freedom from contradiction in the relations into which it entered. The bases of the calculus were then defined formally in terms only of number and infinite aggregates, with no corroboration through an appeal to the world of experience either possible or necessary. This formalizing tendency has not, however, been everywhere accepted. Even mathematicians have not always been in sympathy with the movement. Hermite, whose favorite idea was to compare mathematics with the natural sciences, was horrified by Cantor's work, which transcended human experience.2 Du Bois-Reymond similarly opposed the formal definitions which were made basic in the calculus and wished instead to define number in terms of geometric magnitude, much as Cauchy had done implicitly,3 thus retaining intuition as a guide. More thoroughgoing intuitionists like Brouwer attempt ' See Poincaré, "L'Avenir des mathématiques," p. 939. 3 Pringsheim, "Nombres irrationnels et notion de limite," pp. 153 ff. See also Jourdain, "The Development of the Theory of Transfinite Numbers," 1913-14, pp. 1, 9-10.

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to visualize a fusion of the continuous and the discrete, somewhat as Plato had. 4 The mathematician Kronecker opposed the work of Dedekind and Cantor, not because of its formalism, but because he thought it unnatural. These investigators had "constructed" numbers which Kronecker felt could have no existence, and he proposed instead to base everything on equations involving integers alone, a view which has not been generally shared by other mathematicians. 5 A number of scientists and philosophers have naturally been even more hesitant than these mathematicians in giving up experience and intuition in connection with the calculus. Thoroughgoing empiricists and idealistic philosophers in particular have sought, since the time of Newton and Leibniz to read into the calculus a significance beyond t h a t of a formal postulational system. Newton had considered the calculus as a scientific description of the generation of magnitudes, and Leibniz had viewed it as a metaphysical explanation of such generation. The formalism of the nineteenth century took from the calculus any such preconceptions, leaving only the bare symbolic relationships between abstract mathematical entities. Nevertheless, traces of the old scientific and metaphysical tendencies remained. Lord Kelvin, who considered mathematics the etherealization of corner mon sense, once exclaimed, when he had asked what — represented and had received the answer

li* im AI->O

f\x — : " T h a t ' s what Todhunter would A T

say. Does nobody know t h a t it represents a velocity?" 6 His friend Helmholtz showed a similar tendency. In his famous essay, Ueber die Erhallung der Kraft, he regarded a surface as the sum of lines, 7 much as had Cavalieri in his Exercitationes just two centuries earlier. In another connection he asserted that incommensurable relations may occur in real objects, but t h a t in numbers they can never be repre4 See Helmholtz, Counting and Measuring, pp. xxii-xxiv; also Brouwer, "Intuitionism and Formalism," and Simon, "Historische Bemerkungen iiber das Continuum." 6 See Couturat, De I'infini mathimatique, pp. 603 ff.; cf. also Pierpont, "Mathematical Rigor," pp. 38-40; Pringsheim, "Nombres irratiormels et notion de limite," pp. 158-63; Jourdain, "The Development of the Theory of Transfinite Numbers," 1913-14, pp. 2-8. • Hart, Makers of Science, pp. 278 -79; Felix Klein, Entivicklung dcr Mathcmatik im 19. Jahrhunderl, I, 238. ' Ueber die Erhallung der Kraft, p. 14.

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sen ted with exactness." Mach also felt strongly the empirical origin of mathematics and held with Aristotle that geometric concepts are the product of idealization of physical experiences of spa«ce.' In conformity with this view, he felt that some form of geometrical meaning had necessarily to be given to the number i.10 In this respect he is in agreement with a number of present-day scientists, who feel that —1 simply "forms a part of various ingenious devices for handling otherwise intractable situations." 11 The attitudes of Helmholtz and Mach are representative of the influence in science of the positive philosophy of the nineteenth century. Positivistic and materialistic thought were slow to accept the changed mathematical view and insisted that the calculus be interpreted in terms of velocities and actual intervals, corresponding to the data of experience and of ordinary algebra. Comte recognized that mathematics is not "the science of magnitudes," 12 but he did not rise to the formal view of Cauchy. In accordance with the empirical and pragmatic attitude of Carnot, he regarded the methods of Newton, Leibniz, and Lagrange as fundamentally identical. However, because the differential calculus gave no clear conception of the infinitely small and because the method of limits apparently separated the fields of ordinary and transcendental analysis, he felt that the method of Lagrange was to be favored. 13 More strongly expressed are the views of Diihring, who, in 1872, in his classic Kritische Geschichte der allgemeinen Principien der Mechanik, indulged in a polemic against Gauss, Cauchy, and others who would deny the absolute truth of geometry, and who would introduce into mathematics such figments of the imagination as imaginary numbers, non-Euclidean geometry, and limits!14 Marxian materialists will not grant mathematics the independence of experience necessary for its proper development. 16 Such denial makes impossible the concept of the derivative and the scientific description of motion in terms thereof. The mathematical infinity is, in accord with * Helmholtz, Counting and Measuring, p. 26. 9 Mach, Space and Geometry, p. 94; cf. also p. 67. See also Strong, Procedures and Metaphysics, p. 232. 10 11 Space and Geometry, p. 104, n. Heyl, "The Skeptical Physicist," p. 228. 12 u Comte, The Philosophy of Mathematics, p. 18. Ibid., pp. 110-17. 14 Diihring, Kritische Geschichte der allgemeinen Principien der Mechanik, pp. 475 ff., 529 ff. " Engels, Herr Exigen Dühring's Revolution in Science, p. 47.

308

Conclusion

this view, a contradiction of the "tautology" t h a t the whole is greater than any of its parts." If a number of philosophers were led by excessive realism to reject much of the mathematics of the nineteenth century, idealistic philosophers, following K a n t , were likewise unwilling to accept the bare formalism of Cauchy and Weierstrass in the realm of the calculus. The differential had been defined by Cauchy, not as a fixed quantity, but as a variable, and Weierstrass had shown t h a t the continuous variable depends only upon the static notion of sets of elements. Idealists attempted, nevertheless, to interpret the differential as having an intensive quality resembling the potentiality of Aristotle, the impetus of the Scholastics, the conatus of Hobbes, or the inertia of modern science. They wished to view the continuum, not in terms of the discreteness of Cantor and Dedekind, but as an unanalyzable concept in the form of a metaphysical reality which is intuitively perceived. The differential calculus was regarded as possessing a "positive" meaning as the generator of the continuum, as opposed to the "negation" of the limit concept. 17 As Hegel expressed it, the derivative represented the "becoming" of magnitudes, 18 as opposed to the integral, or the "has become." Materialistic and idealistic philosophies have both failed to appreciate the nature of mathematics, as accepted a t the present time. Mathematics is neither a description of nature nor an explanation of its operation; it is not concerned with physical motion or with the metaphysical generation of quantities. It is merely the symbolic logic of possible relations, 19 and as such is concerned with neither approximate nor absolute truth, but only with hypothetical truth. T h a t is, mathematics determines what conclusions will follow logically from given premises. The conjunction of mathematics and philosophy, or of mathematics and science, is frequently of great service in suggesting new problems and points of view. "Ibid., pp. 48-49, 62; cf. also Bois, "Le Finitisme de Duhring," p. 95. " See Kant, Sdmmtliche Werke, XI (Part I), 270-71; cf. also II, 140-49 and passim. See also Cohen, Die Princip der Infinitesimalmethode und seine Geschichte, passim; Simon, "Zur Geschichte und Philosophie der Differentialrechnung," p. 128; Vivanti, "Note sur l'histoire de 1'infiniment petit," pp. 1 ff.; Freyer, Stiidien zur Metaphysik der Dijfereniioirechnung, pp. 23 ff.; Lasswitz, Geschichte der Atomistik, I, 201. u Klein, Elementary Mathematics from an Advanced Standpoint, p. 217. Cohen, M. R., Reason and Nature, pp. 171-205.

Conclusion

309

Nevertheless, in the final rigorous formulation and elaboration of such concepts as have been introduced, mathematics must necessarily be unprejudiced by any irrelevant elements in the experiences from which they have arisen.*0 Any attempt to restrict the freedom of choice of its postulates and definitions is predicated on the assumption that a given preconceived notion of the nature of the relationships involved is necessarily valid. The calculus is without doubt the greatest aid we have to the discovery and appreciation of physical truth; but the basis for this success is in all probability to be found in the fact that the concepts involved were gradually emancipated from the qualitative preconceptions which result from our experiences of variability and multiplicity. Greek philosophy had attempted to separate and contrast the qualitative and the quantitative, but the later medieval and early modern period associated them through geometric representation. Even a quantitative explanation is subject to sensory notions of size, length, duration, and so forth, so that greater independence was achieved in the nineteenth century by basing the calculus upon ordinal considerations only. The history of the concepts of the calculus shows that the explanation of the qualitative is to be made through the quantitative, and the latter is in turn to be explained through the ordinal, perhaps the most fundamental notion in mathematics. As the sensations of motion and discreteness led to the abstract notions of the calculus, so may sensory experience continue thus to suggest problems for the mathematician, and so may he in turn be free to reduce these to the basic formal logical relationships involved. Thus only may be fully appreciated the twofold aspect of mathematics: as the language of a descriptive interpretation of the relationships discovered in natural phenomena, and as a syllogistic elaboration of arbitrary premises. 50

Cf. Poincaré, Foundations of Sciente, pp. 28-29, 46, 65, 428.

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Index Abel, N. H., 280 Académie des Sciences, 241, 242, 263 Acceleration, Greek astronomy lacked concept of, 72; uniform, 82 ff., 113ff., 130, 165 Achilles, 24-25, 116, 138, 140, 295; argument in the, 24«; refuted by Aristotle, 44 Acta eruditorum, 207, 208, 214, 221, 238 A est unum calidum, 87 Agardb, C. A., Essai sur la métaphysique du calcul intégral. 264 Albert of Saxony, 66, 68f., 74 Alembert, Jean le Rond d', 237; concept of differentials, 246, 275; interpretation of Newton's prime and ultimate ratio, 247; quoted, 248; limit concept, 249, 250, 253, 257, 265, 271, 272; definition of the tangent, 251; concept of infinitesimal, 274; concept of infinity, 275 Algebra, Arabic development, 2, 56, 60, 633., 97, 120, 154; contribution of Hindus, 63; symbols for quantities, 98; use of, avoided in seventeenth century, 302 Algebraic formalism fostered by Euler and Lagrange, 305 Alhazen (Ibn al-Haitham), 63 Ampère, A. M., theory of functions, 282 Anaxagoras, 40 Anaximander, 28, 40 Angle of contact, 22, 173, 174, 212 Antiphon the Sophist, 32 Apeiron, 28 Apollonius, influence in development of analytic geometry, 187 Arabs, algebra adopted from, 2; extended work of Archimedes, 56, 120; algebraic development, 60, 63 ff., 97, 154 Arbogast, L. F. A., Du calcul des dérivations, 263 Archimedes, 25, 29, 38, 39, 48, 49-60, 64, 65, 70, 101, 105, 112, 299, Method, 21, 48-51, 59, 99, 125, 139, 159; axiom of, 33, 173; use of infinitely large and infinitely small, 48ff., 90; quadrature of the parabola, 49-53, 102, 124; volume of conoid, 53-55; spiral, 5558, 133; determination of tangents, 56ff.;

work on centers of gravity, 60; use of series, 76f., 120; mensurational work, 89, 94, 107; translation and publication of works, 94, 96, 98; modifications of method of, Chap. IV; use of doctrine of instantaneous velocity, 56 ff., 130; inequalities, 159 Area, concept of, 9, 31, 34, 58 f., 62, 105, 123; application of, 18, 19, 32, 36, 96 Aristotle, 1, 20, 22,26,30,46, 57, 60,65, 66, 67, 70, 72, 79, 112, 299, 303; opposed to Plato's concept of number, 27; On the Pythagoreans, 37; dependence upon sensory perception, 37ff.; considered mathematics an idealized abstraction from natural science, 37-38, 303, 307; quoted, 38, 41, 42, 43, 46; concept of the infinite, 38ff., 69, 70, 102, 117, 151, 275, 296; concept of number, 40-41, 151, 15455; Physica, 41, 43, 65, 78; theory of continuity, 41-42, 66-67, 277, 291; concept of motion, 42-45, 71 ff., 176, 178, 233; denial of instantaneous velocity, 43, 73, 82; logic of, 45, 46, 47, 61; sixteenthcentury opposition to Aristotelianism, 96; views triumphed in method of exhaustion and geometry of Euclid, 45 ff., 304 Arithmetic triangle, see Pascal's triangle Arrow, argument in, 24«, 295; refuted by Aristotle, 44 Assemblages, see Infinite aggregates Athens, 26 Atomic doctrine, 21, 23, 38, 42, 66, 67, 84, 92, 180, 181 Atomism, mathematical, 22, 28, 50, 82, 115, 125, 134, 176, 188 Average velocity, 6, 8, 75, 83, 113 ff. Babylonians, mathematical knowledge, 1, 14ff., 62, 303; astronomical knowledge, 15 Bacon, Roger, 64; Opus majus, 66; concept of motion, 72 Barrow, Isaac, 58, 133, 209, 299, 302, 304; anticipated invention of the calculus, 164; opposed arithmetization of mathematics, 175; advocated classic concept of number

338

Index

Barrow, Isaac (Continued) and geometry, 179; distrust of algebraic methods, 180; concepts involved in his geometry, 181; quoted, 182; Geometrical Lectures, 182, 185, 189, 202; method of tangents, 182-86, 192-93, 203, 210, 214; influence in development of the calculus, 187; concept of time, 194, 233, 295; differential triangle, 183, 203 Bayle, Pierre, 217, 224 Beaugrand, Jean de, 120 Bede, 66 Berkeley, George, 156, 257 f. ; criticism of Newton, 224ff.; The Analyst, 224-29, 232, 233, 235; Essay towards a New Theory of Vision, 224; influence of empirical tendency, 227, 304; A Defence of Freethinking in Mathematics, 228; Berkeley - Jurin - Robins controversy, 229 ff., 246, 263 Berlin Academy, 254 Bernoulli, James, 153, 207, 220, 238; concept of infinitesimal, 239 Bernoulli, John, 207, 211, 212, 220, 238, 246; concept of infinitesimal, 239, 240; treatment of infinite and infinitesimal, 242; concept of the integral, 278 Bhaskara, 62, 192 Binomial theorem, 119, 148, 190, 191, 194, 207, 225 Blasius of Parma (Biagio Pelicani), 84, 88, 89 Boethius, Geometry, 64 Bolzano, Bernhard, 268, 284; concept of infinity, 116, 270, 274f., 288, 296, 297; Rein analytischer Beweis des Lehrsatzes . . ., 266; limit concept, 269; need for definitions of convergence of series, 270, 280; theory of functions, 276, 282; definition of continuity, 277, 279 Bombelli, Raphael, 97 Boyle, Robert, 2 Bradwardine, Thomas, 66ff., 97, 116; Geometria speculativa, and Tractatus de continua, 66; quoted, 67; Liber de proportionibus, 69, 74 Brahmagupta, 62 Brasseur, J. B. "Exposition nouvelle des principes de calcul différentiel," 264 Brouwer, L. E. J., 28, 67, 305; see also Intuitionism Bruno, Giordano, 107 Bryson of Heraclea, 32, 33, 300

Buffon, Comte de, 246, 247 Buquoy, Georg von, Eine neue Methode für den InfinitesimaJkalkui, 264 Buridan, Jean, 72, 113; doctrine of impetus (or inertia), 177 Burley, Walter, 66, 74 Calculator, see Suiseth, Richard Cambridge Platonists, 180 Cantor, Georg, 33, 271, 284; sought definition of irrational number, 288; "Über die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen," 289; Mengenlehre, 296, 297; concept of the infinite, 296, 297, 298; continuum of, 296ff.,300 Cantor-Dedekind axion, 291-92 Capella, Martianus, 66 Cardan, Jerome, 97, 118; De subtilitaie, 88 Cameades, 38 Carnot, L. N. M., 226; doctrine of perfect and imperfect equations, 158; Réflexions sur la métaphysique du calcul infinitésimal, 257; concept of infinitesimal, 258, 283 Cauchy, A. L., 111, 123, 268; differential defined in terms of the derivative, 210, 275, 276; theorem of, 270, 281, 282, 289; Cours d'analyse, 271; excerpt, 272, 281; Résumé des leçons sur le calcul infinitésimal, 271; Leçons sur le calcul différentiel, 271; definition of limit, 272, 287; ideas of variable, function, limit, and orders of infinity, 212 ft., 281; concept of infinitesimal, 273; definition of continuity, 277, 279 Cavalieri, Bonavcntura, 28,48, 56,109, 151, 160, 204, 299; theorem of, 22, 118; Geometría indivisibilibus, 111, 112, 117, 119, 121, 122, 123; forces molding thought of, 112, 117 ft.; Exercitaliones geometricae sex, 117, 135, 306; role in development of infinitesimal methods, 121,123; method of indivisibles, 125, 141, 142, 150, 151, 170, 181, 195, 227 Characteristic triangle, see Differential triangle Circle, conception of as a limit, 32, 231, 272 Collins, John, 185 Commandino, Federigo, 99, 101, 112, 159; Liber de centro graviiatis solidorum, 94

Index Commercium epistolicum, 221, 222 Compensation of errors, 226, 245, 251, 257 Comte, Auguste, 307 Conatus, Hobbes's concept of, 43,178, 179, 212, 308 Condorcet, N. C. de, 263 Continuity, concept of, 3, 4, 25, 26, 29, 35, 37, 38, 41, 44, 47, 51, 62, 63, 65-68, 116, 151, 152, 155, 179, 181, 184, 226, 229, 246.. 267-70, 279, 282, 288, 291, 292, 294, 295; Aristotle's conception, 41-45, 304; medieval conception, 65 ff., 304; law of, 93, 110, 116, 172, 178, 217-22 passim, 244, 245, 256, 258, 268, 277,281, 283, 294, 299; Leibniz's doctrine of, 217, 268, 277, 291; definition of, 268, 277, 279, 286, 294, 298 Continuum, 3, 4, 13, 16, 17, 20, 42, 43, 45, 60, 66, 67, 115, 116, 179, 181, 217, 218, 256, 270, 291; generated by flowing of apeiron, 28; mathematical concept of, 35, 47, 92, 227; attempted link with indivisibles, 91; Barrow's treatment of, 180; views of Cantor imputed to earlier investigators, 300; as an unanalyzable concept, 308 Convergence of series, 174, 207, 253, 264, 270, 280 ff. Coumot, A. A., Traité élémentaire de la théorie des fonctions et du calcul infinitésimal, 283 Crelle's Journal, see Journal für die Reine und Angewandte Mathematik Cusa, Nicholas of, see Nicholas of Cusa Cycloid, 151, 167 Dedekind, Richard, basis of definition of infinite assemblages, 271; Stetigkeit und die Irraitonalzahlen, 289; theory of irrational numbers, 290ff.; concept of the infinite, 296 Dedekind Cut, 33, 42, 292, 294, 300 Definite integral, see Integral, definite De lineis insecabilibus, 39 Democritus, 21, 22, 26, 28, 30, 37, 39, 48, 66, 71, 91, 108; see also Atomic doctrine; Atomism, mathematical Derivative, concept of, 6, 13, 22, 28, 40, 43,47, 59, 61, 63, 79, 80, 82, 85, 111, 157, 179, 184, 185, 186, 191, 196, 251, 255, 259, 262, 277, 282, 295, 298, 300, 308;

339

defined, 3, 4, 5, 7, 8, 11, 12, 269, 275, 293, 294; differential defined in terms of, 210, 275, 276; origin of word, 252-54; continuous functions and, 269, 270, 279 Derived functions, Lagrange's method of, 251-55, 259-65, 267, 270, 272, 275, 278; see also Derivative Descartes, René, 2, 29, 58, 88, 133, 165, 238; Géométrie, 154, 166, 208; concept of variability, 154, 155; method for normal to a curve, 158; infinitesimal concept, 165, 168, 193; method of tangents, 16668; concept of instantaneous center of rotation, 167; notion of instantaneous velocity, 168, 177; Discours de la méthode, 168; analytic geometry, 154, 165, 179, 187; number concept, 242; method of undetermined coefficients, 166, 259 Dichotomy, 116, 295; argument in, 24»; argument refuted by Aristotle, 44 Difference, see Differential Differences, method of, see Differentials, method of Differential coefficient, 276; origin of term, 253, 265; see also Derivative Differential quotient, 263, 275; integral the inverse of, 283; see also Derivative Differentials, 111, 123, 150, 173, 179, 187223, 224, 278; defined, 12, 210, 259, 275, 276, 308; method of. 59, 122, 145, 178, 185, 187-223, 224, 226, 234, 236ff., 243, 244ff., 251, 253, 266, 267, 277, 294, 299, 300, 302; origin of word, 205-6 Differential triangle, 152,163,182,203, 218, 242 Diocles, curve defined by motion, 71 Diogenes Laertius, 22 Diophantine equations, 72 Diophantus, 63; Arithmetic, 60 Distance, represented by area under velocity-time curve, 82-84, 113-15, 126, 130-31, 134, 180 Divergent series, 207, 241, 245, 246, 265, 280 Du Bois-Reymond, Paul, 305 Diihring, Eugen, Kritische Geschichte der allgemeinen Principien der Mechanik, 307 Duns Scotus, John, 66, 74 Dynamics, 45, 178; Galilean, 72, 82, 83, 113-15, 130, 177-78; medieval, 72ff., 82 ff.

340

Index

Egyptian mathematics, 1, 14 ff., 21, 303 Eleatics, 23 Encyclopedia Britannica, excerpt on fluxion, 232 Epicurus, 22, 91 Eratosthenes, 48 Euclid, 1, 2, 22, 27, 30, 48, 61, 64, 90, 173; Elements, 20, 45 ff., 64; debt to Eudoxus, 30; quoted, 31, 33, 240; method of exhaustion, 34, 41; logic, 46ff.; view of number, 47, 221; banned concepts which gave birth to the calculus, 47-48, 301; concept of tangent, 57 Eudoxus, 30ff., 37, 45, 48, 299; method of exhaustion, 30, 31, 34, 41, 46, 96; definition of proportion, 31, 47, 300 Euler, Leonhard, 167, 224, 226, 234, 250; concept of differentials, 150, 253; function concept, 243, 272, 275; view of the infinite, 245, 249; divergent series, 246, 280; concept of the integral, 278 Evanescent quantities, 216, 228 Exhaustion, method of, 4, 9, 26, 30, 31, 32 ff., 36, 41,46,48, 51 ff., 62, 95, 99, 100, 104,106,109,123,124, 128,133,136,150, 171, 197, 215, 255, 258, 259, 271, 300, 304; origin of word, 34, 136, 140; concept of infinity substituted for method of, 142

Faille, Jean-Charles della, 138 Fermat, Pierre de, 48, 56, 120, 121, 125, 127,140,142,154,164,168,169,175,187, 208, 209 f., 299, 300; method of maxima and minima, 111, 155-59, 163, 164, 18586, 202; method of tangents, 157, 162 g., 167, 175, 183ff., 192, 202; quadratures, 159-64, 202; use of infinite progressions, 160-61, 190; concept of definite integral, 161-62, 173; rectification of curves, 162; differential triangle, 163, 203; Opera varia, 164 Ferrari, Lodovico, 97 Fluent, 79, 187-223 Fluxions, 79, 173, 187-223, 224, 226, 232, 250, 260, 278; method of, 58,59,122,129, 178, 187-223, 224, 225,226, 229, 236, 237, 238, 243, 246,247, 251, 253, 263, 265, 266, 267, 294, 299; views of Robins, Maclaurin, Taylor and Simpson, 232 ff. Fonctions dérivées, see Lagrange, J. L., method of derived functions

Fontenelle, Bernard de, 214, 241, 249; treatment of infinite and infinitesimal, 242; Élémens de la géométrie de l'infini, 242 Formalism, see Mathematical formalism Forms, latitude of, 73ff., 82, 85, 86, 87, 88, 126; graphical representation, 81-84, 111 Fourier, J. B., function concept, 276 Function concept, 55, 56, 58, 60, 94, 98, 156, 184, 186, 196, 220, 221, 236, 237, 240, 243, 259, 272, 273, 276 Fundamental theorem of the calculus, 10, 11, 164, 181, 184, 187, 191, 194, 196, 203, 204 ff., 279 Galileo, 2, 29, 70, 121, 177, 180, 299; concept of the infinite, 70, 115ff., 193, 270; dynamics, 72, 82, 83, 89, 113ff., 133; recognition of law of uniform acceleration, 82, 85, 113ff., 130; conccpt of area, 84, 134; Two New Sciences, 85, 112, 113, 115; forces molding thought of, 1122.; idea of impetus, 113, 130, 177; concept of the infinitely small, 115ff., 195; doctrine of continuity, 116, 291 Gassendi, Pierre, 224 Gauss, K. F., 98, 268, 280 Geometry, Greek, see Greeks Gerdil, Hyacinth Sigismund, 250 Girard, Albert, 104 Goethals, Henry, 66, 74 Grandi, Guido, 218; interest in differential calculus, 238; concept of infinitesimal, 241 Graphical representation of variables, 80 ff., I l l , 113ff., 190, 288, 309 Gravity, centers of, 60, 93, 94, 99 ff., 104, 139, 145, 158, 163, 166 Greeks, search for universals, 8, 14; development of mathematics and science, 16ff.; geometry, 17, 26, 46, 65; concept of number, 18, 29, 41, 43; position of Zeno's paradoxes in thought of, 25; concept of length, area, and volume, 31 ff.; concept of proportion, 31 ff.; method of exhaustion, 33 ff., 35, 100, 136, 197, 271, 300; concept of motion, 42 ff., 71, 132; skcptics, 46, 214; abandonment of attempt to associate numbers with all geometric magnitudes, 20, 62; mathematics in Middle Ages, 64; concept of instantaneous velocity, 43, 57, 82

Index Gregory, James, 192, 280; Vera circuit et hyperbolae quodroiura, 174; arithmetical and analytical work, 175; quadratures, 175; use of Fermat's tangent method, 175, 183 Gregory of Rimini, 66, 69 Gregory of St. Vincent, 162, 202; Opus geometricum, 135, 160, 169; "ductus plani in planum," 135, 145; view of nature of infinitesimals, 135 S.; use of word "exhaust," 136; limit concept, 137, 152, 169, 282; geometric progressions, 137-38,174,190; influence, disciples, 138; method of indivisibles, 141 Grosseteste, Robert, 66 Guldin, Paul, 121, 138 Guldin's Rule, see Pappus, theorem

Hadamard, J., 276 Halley, Edmund, 222, 224 Ilankel, Hermann, 271 Harmonic triangle, 204 Harriot, Thomas, 169 Hegel, G. W. F., 308 Heine, Eduard, 290; limit concept, 287; "Die Elemente der Funktionenlehre," 288 Helmholtz, Hermann von, 294, 307; Die Erhaltung der Kraft, 306 Hentisbery, William of, 84, 88, 96, 113, 177 Heraclitus, 25, 71 Hermite, Charles, 305 Heron of Alexandria, 59, 115 Ileterogenea, 140, 152 Heuraet, Heinrich van, rectification of curves, 162 Hindus, Pythagoras' debt to, 19; mathematics, 60, 61 ff., 72, 97 Hipparchus, 59, 63 Hippias of Elis, 56, 71 Hippocrates of Chios, 33 Hobbes, Thomas, 212, 227; concept of the conatus, 43, 178, 179, 195, 308; opposed arithmetization of mathematics, 175; concept of number and of the geometrical elements, 176; idea of motion, 178 Homogenea, 140 Horn angle, see Angle of contact Hudde, Johann, rules for tangents and for maxima and minima, 185 f. Humanism, 89

341

Huygens, Christiaan, 153, 208, 213; use of Fermat's tangent method, 183,186; rules for tangents and for ma-rima and minima, 185 f. Ibn al-Haitham, see Alhazen Impetus (inertia), concept of, 43, 72, 130, 177 Incommensurable, see Irrational Indefinite integral, see Integral, indefinite Indivisibles, 11, 13, 22 , 27, 38, 63, 66ff., 84,91,109,116,118S., 123,124,151,173, 197, 227; method of, 50, 56, 92, 117-26, 133,134,139,140,141,142,143 ff., 149 ff., 181 Inertia, see Impetus Infinite aggregates, 13, 25, 68, 70, 115, 249, 257, 268,270, 275, 277, 284,286,290, 291, 293-98; definition, 296 Infinitely small, see Infinitesimal Infinite sequence, 4, 7, 8, 9, 11, 24, 36, 37, 50, 53, 102; see also Infinite series Infinite series, 24,44, 52, 55, 56, 76, 86,116, 126, 137, 140, 142, 152, 154, 173, 190, 192, 199, 216, 221, 226, 231, 245, 254, 256, 264, 265, 267, 268, 270, 276, 280, 281, 282, 284, 286, 289, 290, 302; convergence of, 174, 207, 253, 264, 270, 280 ff. Infinitesimal, 7, 11, 12, 22, 29, 50, 51, 55, 59, 62, 63, 65, 68, 83, 84, 90, 92, 96, 108, 109, 114, 115, 117, 135, 142, 145, 152, 162, 163, 166, 169, 176, 180, 181, 183 ff., 187, 191, 193, 195, 196, 197, 199, 200, 201, 202 ff., 209 ff., 212, 215, 218, 220, 224, 225, 229 ff., 258, 260, 263, 265, 286, 299, 301; concept of, 4, 21, 28, 39, 47, 154ff., 213ff., 239ff., 262, 300 (see also under Cavalieri; Descartes; Pascal); not found before Greek period, 15; Zeno's dictum against, 23; Democritean view, 30; Aristotelian view, 38ff.; as an intensive quantity, 43, 178f., 262; Archimedes' view, 49ff.; medieval views, 66ff.; potential existence, 67; view of Nicholas of Cusa, 90ff.; Kepler's view, 108ff.; Galileo's view, 112flf.;view of Gregory of St. Vincent, 135ff.; Tacquet's view, 139ff.; Roberval's view, 141 ff.; Pascal's view, 149ff.; Fermat's view, 154ff.; Wallis' view of, 170ff.; philosophers reluctant to abandon, 179; Barrow's

342

Index

Infinitesimal (Continued) view, 180ff.; Newton's view, 193ff.; Leibniz's view, 209ff.; confused with fluxions, 223; Jurin's view, 229; Robins' disavowal of, 230; views of John and James Bernoulli, 238II.; views of Wolff and Grandi, 240-41 ; view of Fontenelle, 241-42; Euler's view, 244-46; definition, 248, 273; D'Alembert's view, 248f.; views of Kästner and Lagrange, 250 ff.; Carnot's view, 258 f.; views of Bolzano and Cauchy, 270 ff. Infinitesimals, fixed, see Indivisibles Infinity, 2, 6, 22, 38, 41, 44, 51, 52, 56, 63, 65, 66, 70, 76, 77, 90, 91, 92, 108, 121, 143, 151, 152, 172, 213, 218, 239ff., 267; concept of, 25, 39, 47, 53, 115 ff., 226, 268, 296, 302, 304; Aristotle's view, 3841, 68; actual and potential, 40, 41, 68, 69, 77, 102, 117, 151; banned from Greek mathematics, 46, 47 , 96, 142, 301; Scholastic view, 68ff.; Galileo's view, 70, 115,193, 270; mathematical concept, 116, 170-71, 227, 270-71, 274-75, 284, 296ff.; Bolzano's view, 116, 270-71, 296; symbols used to represent, 170, 297; orders of, 245, 274; Cauchy's view, 274-75, 284, 296; view of Dedekind and Cantor, 296ff. Inscribed and circumscribed figures, proposition of Luca Valerio, 124, 128, 137 Instantaneous center of rotation, 167 Instantaneous direction, 57 f., 132,134,146, 189, 212 Instantaneous rate of change, 78, 81, 82, 83 Instantaneous velocity, 6, 7, 8, 43, 44, 47, 59, 73, 82, 84, 115, 116, 130ff., 168, 178, 179, 180, 191, 193ff., 200, 221, 227, 232, 233, 295; quantitative treatment, 73 ff., 82ff., 113ff., 130ff., 177; medieval view, 73 ff.; Barrow's treatment, 180; Newton's view, 194ff.; views of Robins, Maclaurin, Taylor, and Simpson, 234 Instants, 92 Integral, 22, 40, 80, 109, 123, 161, 187-223 , 293, 294, 298; defined, 3, 5, 11, 12, 206, 239, 278, 279, 294; concepts of, 6, 13, 47, 56, 61, 63, 184, 185, 295, 308; origin of word, 67, 205, 206 Integral, definite, 8, 50, 56, 120, 121, 123, 127, 128, 143, 144, 146, 149, 152, 159, 161, 191, 206, 278; defined, 9, 10, 55, 279; concept of, 142, 173 Integral, indefinite, 191,206,278; defined, 11

Integral of Lebesque, 280 Integration, by parts, 152, 163 Intensity, average, 74-78 Intensive magnitude, 28, 43, 177, 178, 179, 262, 308 Intuition, 5; Euclid's Elements based on, 47; of uniform rate of change, 78 Intuitionism, mathematical, 3, 28, 67, 209, 305 Ionians, 18, 21; empirical science the basis of mathematics of, 303 Irrational and incommensurable, 18 ff., 20, 31 ff., 35, 37, 61 ff., 66, 96, 97, 107, 121, 173, 174, 176, 179, 190, 256, 281, 282, 284, 285, 288-94, 296, 300, 306 Isidore of Seville, 66 James of Forlì, 74 John XXI, pope (Petrus Hispanus), Summulae logicales, 68 Jordanus Nemorarius, 64, 96, 98, 173 Journal des savants, 238 Journal für die Reine und Angewandte Mathematik, 288 Jurin, James, Geometry No Friend to Infidelity, 228; The Minute Mathematician, 228; attitude toward infinitesimal, 229; controversy with Berkeley and Robins, 229ff., 246, 263 Kant, Immanuel, 2, 3, 261, 264, 308 Kästner, A. G., 254; Anfangsgründe der Analysis des Unendlichen, 250 Kelvin, William Thomson, baron, quoted, 306 Kepler, Johann, 2, 48, 91, 94, 156, 299; influenced by Nicholas of Cusa, 93; speculative tendency, 106; Mysterium cosmographicum, 107; work on curvilinear mensuration, 108; Nova stereometria doliorum, 108, 110, 111, 119; Astronomia nova, 109; theory of maxima and minima, 110; static approach to calculus, 111 ; forces molding thought of, 112; view of infinity, 117 Kronecker, Leopold, opposed work of Dedekind and Cantor, 306 La Chapelle, de, Institutions de géométrie, 247 Lacroix, S. F., 253; Traité du calcul différentiel et du calcul intégral, 264; Traité éli-

Index mentaire, 265; limit concept, 271, 272; concept of the integral, 278; employment of divergent series, 280 Lagrange, J. L., 224, 226, 251; quoted, 165, 252; method of derived functions, 251-55, 259-65, 267, 270, 272, 275, 278; concept of convergent series, 253, 270, 280; Théorie des fonctions analytiques, 260-61, 263 Landen, John, The Residual Analysis, 236 Laplace, Pierre Simon, Marquis de, 224 Latitude of forms, see Forms, latitude of Law of continuity, see Continuity Law of uniformly difform variation, see Uniform acceleration Lebesque, integral of, 280 Leibniz, Gottfried Wilhelm von, 4, 8, 10, 13, 29, 47, 48, 59, 67, 82, 145, 153, 162, 187-223, 230, 237, 287, 299, 300; infinitesimal method, 28, 158, 210, 213, 219, 243, 265, 273, 274; reference to Calculator, 88; sought basis of calculus in generation of magnitudes, 94; differential method, 122, 178, 219, 239, 244, 247, 275, 294; views influenced by Pascal, 150; concept of integral as totality, 173; differential triangle, 182, 203; development of algorithmic procedure, 185, 188, 202; definition of the integral, 206, 278; "A New Method for Maxima and Minima," 207; definition of first-order differentials, 210; Historia et origo calculi dijferentialis, 215; quoted, 217, 218; Thiodicie, 218; view of the infinite, 219, 297; priority claims of Newton and, 188, 221, 222, 246; law of continuity (see under Continuity) Leibniz's Rule, 252 Length, concept of, 19, 31, 58 f. Leonardo da Vinci, 2, 88; influenced by Scholastic thought, 92; center of gravity of tetrahedron, 93; use of doctrine of instantaneous velocity, 130; idea of impetus familiar to, 177 Leonardo of Pisa, 65, 71, 97; Liber abaci, 64 Leucippus, 66 Lever, law of the, 49 L'Hospital, Marquis de, 226; Analyse des infiniments petits, 238, 241 L'Huilier, Simon, Exposition élémentaire des principes des calculs supérieurs, 255; definition of differential quotient, 255; limit concept, 257, 265, 271, 272; view of differentials, 275; view of the integral, 278

343

Limit, 15, 25, 34, 37,50,51, 52, 53,62,109, 121,126,158,174,180,197,198,200,201, 211, 213, 214, 218, 220, 226, 255, 260; definition, 7, 8, 36, 272, 287; derivative based on the idea of, 7, 8, 279, 293, 294; concept of, 24, 27, 30, 32, 43, 53 ff., 58, 73, 77, 79, 85, 87, 91, 98, 102 ff., 106,114, 117,123,124,132,134,137,142,153,156, 167,169,173,176,179,180,181,195, 198, 216, 218, 222, 228, 233, 235, 244 ff., 262, 266,271 ff., 281,282,284,286ff., 295,298; method of, 100 ff., 117,146,150,157, 229, 240, 263, 264, 267, 284; doctrine of, 140; controversy between Robins and Jurin, 230; calculus interpreted in terms of, 247ff., 254ff., 267ff.; involved definition of number, 290, 296 Lines, incommensurability of, 19, 20; as velocities or moments, 81 ff., 114,115,117 Lucretius, 68

Mach, Ernst, view of continuity, 291, 294; geometric concepts, 307 Maclaurin, Colin, 82, 234-35; concepts of time and instantaneous velocity, 233; Treatise of Fluxions, 233, 260 Major, John, 85 Marsilius of Inghen, 84 Mathematical atomism, see Atomism, mathematical Mathematical formalism, 3, 67, 209, 243, 246, 253, 272, 283, 284, 286, 305 Mathematical Gazette, 276 Mathemalische Annalen, 289 Maurolycus, Franciscus, 159 Maxima and minima, 16, 85, 110, 155-59, 163, 164, 185, 192, 208 Mengenlehre, 296, 297; see also Infinite aggregates Méray, Charles, Nouveau précis d'analyse infinitésimale, 288, 289; "Remarques sur la nature des quantités définies," 289 Mersenne, Marin, 165, 166 Middle Ages, 2, 60, 61-95; speculations on infinite, infinitesimal, and continuity, 94; reaction against work of the, 96 Miscellanea Taurinensia, 254 Moments, 150, 178, 191, 193, 194, 195, 198, 212, 221, 236; lines as, 114, 115, 117; Newton's use of, 122, 229, 235; confusion between fluxions and, 224, 232 Monad, 23, 28

344

Index

Montucla, Etienne, 151 Motion, nature of, 6-7, 42, 294-95; instantaneous direction of, 57, 130ff., 14647, 166-67; uniformly accelerated, 82 ff., 11311, 130, 165; graphical representation of, 80 ff. Napier, John, 122 Neil, William, rectification of curves, 162 Neoplatonism, 45, 177 Newton, Sir Isaac, 4, 10, 13, 28, 29, 47, 48, 51, 59, 82, 116, 150, 162, 174, 180, 187223, 236, 268, 277, 278, 299, 300; method of fluxions, 58, 79, 82, 122, 129, 133, 178, 190, 213, 220, 221, 225, 229, 278, 294; view of moments, 122, 235; concept of limit, 145, 271; invention of algorithmic procedures, 185, 188; concept of number, 190, 242; De analyst, 190, 194, 200, 202, 228, 235; use of infinite series, 190; use of infinitely small, and binomial theorem, 191, 194; Methodus fluxionum, 193, 194, 195, 196, 200, 206, 246; definition of fluent and fluxion, 194, 206; De quadrature 195, 196, 197, 200, 201, 202, 205, 206, 221, 222, 225, 228, 247; Principia, 197,198,199, 200,201, 205,220,222,225, 228, 229, 271; concept of infinitesimal, 226, 274; renunciation of infinitesimals, 213, 222; priority claims of Leibniz and, 221, 222, 246; prime and ultimate ratios, 222, 225, 229ff., 254, 259, 287, 294, 300; Berkeley's criticism of, 224ff.; need for clarification of terms used by, 224, 228, 230; view of continuity of time, 194, 295 Nicholas of Cusa, 2, 88, 89 ff., 93, 107, 108, 135; quadrature of the circle, 91; views on infinite and infinitesimal, 92, 93; idea of continuity, 110; influence, 113; idea of impetus familiar to, 177; influenced by Platonic view, 88-90, 107, 304 Nicomachus, 143-44 Nicomedes, 71 Nieuwentijdt, Bernard, 213, 224, 248 Nondifferentiable continuous function, 269, 270, 276, 282, 285, 288 Notations of the calculus, 191, 193-95, 198, 205, 220, 223, 252, 265, 268, 275 Number, concept of, 15, 29, 30, 31, 36, 41, 42, 46, 51, 52, 60, 61, 62, 63, 71, 97, 154, 174, 176, 178, 179, 180, 190, 216, 221, 226, 231, 242, 256, 270, 273, 281, 282, 284,

285, 286, 289-94, 296, 298, 304, 305, 306; Pythagorean view, 18ff.; Greek view, 20, 40-45; Plato's view, 27 Numerals, Hindu-Arabic, 63, 71 Oldenburg, Henry, 196 Oresme, Nicole, 65, 79ff., 84, 96, 177, 299; Tractatus de laiitudinibus formarum, 80, 85, 88; Tractatus de figuraiiont potentiarum et mensurarum, 81; idea of instantaneous velocity, 82ff.; consideration of infinite series, 86; study of maxima and minima, 85-86, 111, 156; concept of uniform acceleration, 82-83, 113, 130; use of geometric representations, 80ff., 125, 154; influence in development of analytic geometry, 81 ff., 154, 165, 187; infinitesimal concept, 84 ff., 114ff., 193 Oughtred, William, 170 Oxford school, 74, 80, 84, 87 Pacioli, Luca, 88; Summa de arithmetica, 64, 97 Pappus of Alexandria, 155; theorem, 60, 63, 108, 139 Parabola, quadrature of the, 49-53, 159 ff. Paracelsus, 177 Parallelogram of velocities, 57, 129-34, 146 Paris School, 80, 84, 87 Parmenides, 23 Pascal, Blaise, 56, 140, 142, 150, 160, 162, 168, 209; concept of infinitesimal, 121, 147, 148, 150, 151, 239; Potestatum numericarum summa, 148; Traité des sinus du quart de cercle, 151, 153, 203; avoided the use of algebra and analytic geometry, 152-53, 302 Pascal, Etienne, 147 Pascal's triangle, 148, 149, 204 Pericles, 26 Petrus Hispanus, see John X X I , pope Physis, 21 Plato, 1, 21, 25, 26, 29, 30, 37, 39,45,46, 48, 89, 90, 91, 93, 291, 299, 303; concept of mathematics, 1, 26 ff., 60, 303, 304; criterion of reality, 28; influence, 89-90, 93, 107 Plutarch, 22; quoted, 23 Poincaré, Henri, 13 Poisson, S. D., Traité de mécanique, 283 Positivism, 5, 307

Index

345

Prime and ultimate ratios, 116,195 ff., 213, Scholastics, 2, 4, 26, 40, 60, 61-95; view of impetus, 43, 72, 177; discussions of the 216ff., 221, 250, 256, 262, 267; probable infinitely large and small, 66ff., 77, 90; origin of term "ultimate ratio," 197; categorematic and syncategorem&tic inNewton's method, 222, 225, 229 ff., 254, finities, 68ff., 115, 170, 296-97; interest 259, 287, 294, 300; set also Number, in dynamics, 72f., 82ff., 133; concept of concept of instantaneous velocity, 73, 82; consideraProblemum austriacum, see Gregory of St. tion of infinite series, 76 ff., 86 ff., 190; Vincent, Opus geometricum geometrical representation of variables, Proclus, 20, 46; quoted, 17 80ff., I l l ; knowledge of uniformly Proportion, concept of, 30 ff., 47 accelerated motion, 82ff.; reaction against Ptolemy, 59 methodology of, 96; influence upon Pythagoras, 17, 18, 89, 299, 303; debt to Galileo, 112 Hindus, 19 Pythagoreans, 1, 18ff., 26, 39; mathemati- Sensory perception, limitations, 6, 37, 38, cal concepts, 17 ff., 18, 40, 176; applica39, 43, 295 tion of areas, 18ff., 32, 62, 96; problem of Servois, F. J., Essai sur un nouveau mode the incommensurable, 18ff., 61 ; influence, d'exposition des principes du calcul dif89, 93, 107, 115, 143 férentiel, 263 Simplicius, 22 Quadratures, 49-53, 91, 104f., 108, 119ff., Simpson, Thomas, view on instantaneous 124ff., 144ff.,149f., 159 ff., 170ff., 191 ff., velocities, 234 Quartic, solved, 97 Skeptics, Greek, 46, 214 Sluse, René François de, 162, 185 Snell, Willebrord, 109; Tiphys Batavus, Ramus, Petras, 96 152 Raphson, Joseph, The History of Fluxions, 222 Sniadecki, J. B., 261 Ratios, see Number concept; Prime and Soto, Dominic, 85 ultimate ratios Spiral, of Archimedes, 55 ff., 133 Rectification of curves, 133, 162, 163 Stade, 295; argument in the, 24»; argument Regiomontanus, 92 refuted by Aristotle, 44 Riemann, G. F. B., 123 Stevin, Simon, 91, 104, 108, 123, 130, 145, Roberval, Giles Persone de, 56, 58,125,133, 152, 299; propositions on centers of 140 ff., 151, 152, 160, 162, 168, 177, 209, gravity, 99ff.; limit idea, 102, 137, 169; 304; concept of infinitesimal, 121, 141 ff.; influence, 141 Traité des indivisibles, 141, 147, 152; Stifel, Michael, 93, 97, 108 method of indivisibles, 141 ff., 150; Strato of Lampsacus, 39 association of numbers and geometrical Suiseth, Richard (Calculator), 83, 88, 96, magnitudes, 142; obscured limit idea 113, 115, 130, 177, 299; Liber calculathrough notion of indivisibles, 144; tionum, 69, 74, 87, 88; concept of variaarithmetizing tendency in quadratures, bility, 74 ff., consideration of infinite 142 ff., 168; method of tangents, 146 series, 76 ff., 86; use of words fluxion and Robervallian lines, 145, 175 fluent, 79, 194 Robins, Benjamin, controversy with Berke- Symbols, for quantities in algebra, 98; used ley and Jurin, 229ff., 246, 263; A Disto represent infinity, 170, 297; used in course Concerning . . . Newton's Methods, calculus, 194, 205, 253 229; limit concept, 230 ff., 271; disavowal of infinitesimals, 230; criticism of Euler, 246; interpretation of Newton's Tacquet, Andreas, 138; Cylindricorum et prime and ultimate ratio, 247 annularium, 139, 151; Arithmeticae theoRolle, Michel, 241 ria et praxis, 140; method of indivisibles, Royal Society, 221 140, 151, 181; limit concept, 140, 152, Russell, Bertrand, 3», 67; definition of real 169, 230; propositions on geometric number, 293 progressions, 140, 174, 190

346

Index

Tangent, Greek concepts, 56 ff.; Archimedes' method, 57 f.; Torricelli's method, 128-33, 146, 157; definition, 129, 132, 166, 174, 249; Roberval's method, 14647; Fermat's method, 157-58, 162-64, 167, 175, 183, 192-93, 202, 203; method of Descartes, 166; Barrow's method, 18286, 192-93 Tartaglia, Nicolo, 94, 97 Taylor, Brook, 235, 244-45; Methodus incremeniorum, 234 Taylor's series, 261, 267 Thales, 1, 16, 17, 28 Theophrastus, 39 Thomas, Alvarus, 85; Liber de triplici motu, 87 Thomson, William, see Kelvin, William Thomson, baron Time, concept of, 180, 194, 233, 295 Todhunter, Isaac, 306 Torricelli, Evangelist», 56, 58, 59, 122, 123 ff., 134, 145, 160, 162, 168, 177, 299, 304; view of infinitesimals, 121; De dimensione parabolae, 124; use of method of indivisibles, 125, 133, 134; De infinitis hyperbolis, 127, 152; method of tangents, 128ff., 146, 157; method of exhaustion, 128, 133; concepts of motion and time, 130ff., 168,177,180; differential triangle, 203 Transfinite number, 298 Triangle, arithmetic, see Pascal's Triangle Triangle, differential, 152, 163, 182, 203, 218, 242; harmonic, 204 Ultimate ratio, see Prime and ultimate ratio Uniform acceleration, 82 ff., 113ff., 130,165 Uniform intensity, concept of, 83 Uniformly difform variation, law of, see Uniform acceleration Valerio, Luca, 104ff., 108, 112, 123, 299; De ceniro gravitaiis solidorum, 104; proposition on inscribed and circumscribed figures, 105, 124, 128, 137; limit concept, 106, 230 Variability, concept of, 4, 24, 25, 42 ff., 47, 51, 55, 56, 60, 71ff.,78ff.,92, 94, 96, 98,

122, 154ff., 167, 176ff., 180, 193ff., 273, 286, 288, 294, 301; see also Graphical representation of variables; and Forms, latitude of Variation, law for uniformly difform, see Uniform acceleration Varignon, Pierre, 241, 280 Velocities, parallelogram of, 57, 129-34, 146; see also Instantaneous velocity Venatorius, 94 Viète, François, 93, 98, 108; influence in development of analytic geometry, 154, 187 Volume, concept of, 31, 34, 58, 123 Wallis, John, 56, 59; treatment of infinite and infinitesimal, 121, 170, 239, 242; Arithmetica infinilorum, 140, 170, 171, 176, 189; applied analytic geometry to quadratures, 168f.; limit concept, 169; De sectionibus conicis traclatus, 170, 171; concept of definite integral, 173; interest in angle of contact, 173; arithmetization criticized, 179; Algebra, 201 Weierstrass, Karl, 31; continuous nondifferentiable function, 270, 282, 288; trend toward formalism, 283; limit concept, 284, 286-88, 300; concept of irrational number, 285-300 passim; concept of the infinite, 288, 296 Whewell, William, 256 William of Hentisbery, see Hentisbery, William William of Occam, 66, 67, 69, 300 Wolff, Christian, 238; concept of infinitesimal, 240; number concept, 243 Wren, Sir Christopher, 162, 169 Wronski, Hoëné, differential method, 261; Réfutation de la théorie des fonctions analytiques, 261», 264; limit concept, 262 Xenocrates, 39 Xenophanes of Colophon, 23 Zeno, 43, 267, 272, 299, 300; paradoxes, 4, 8, 23 ff., 37, 44, 53, 62, 77, 116, 138, 140, 179, 231, 281, 287, 288, 295, 302